advanced optimization and statistical methods in portfolio … · 2014-09-05 · this dissertation...

Advanced Optimization and Statistical

Methods in Portfolio Optimization and

Supply Chain Management

Ümit Saglam

Submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

under the Executive Committee

of the LeBow College of Business

DREXEL UNIVERSITY

2014

© 2014

Ümit Saglam

All Rights Reserved

ABSTRACT

Advanced Optimization and Statistical

Methods in Portfolio Optimization and


Ümit Saglam

This dissertation is on advanced mathematical programming with applications in

portfolio optimization and supply chain management. Specifically, this research

started with modeling and solving large and complex optimization problems with

cone constraints and discrete variables, and then expanded to include problems

with multiple decision perspectives and nonlinear behavior. The original work

and its extensions are motivated by real world business problems.

The first contribution of this dissertation, is to algorithmic work for mixed-

integer second-order cone programming problems (MISOCPs), which is of new

interest to the research community. This dissertation is among the first ones in

the field and seeks to develop a robust and effective approach to solving these prob-

lems. There is a variety of important application areas of this class of problems

ranging from network reliability to data mining, and from finance to operations

management.

This dissertation also contributes to three applications that require the solution

of complex optimization problems. The first two applications arise in portfolio

optimization, and the third application is from supply chain management. In our

first study, we consider both single-period and multi-period portfolio optimization

problems based on the Markowitz (1952) mean/variance framework. We have

also included transaction costs, conditional value-at-risk (CVaR) constraints, and

diversification constraints to approach more realistic scenarios that an investor

should take into account when he is constructing his portfolio. Our second work

proposes the empirical validation of posing the portfolio selection problem as a

Bayesian decision problem dependent on mean, variance and skewness of future

returns by comparing it with traditional mean/variance efficient portfolios. The

last work seeks supply chain coordination under multi-product batch production

and truck shipment scheduling under different shipping policies. These works

present a thorough study of the following research foci: modeling and solution of

large and complex optimization problems, and their applications in supply chain

management and portfolio optimization.

Table of Contents

1 Introduction 1

1.1 Algorithmic Studies: MISOCP . . . . . . . . . . . . . . . . . . . . 2

1.2 Application 1: Portfolio Selection Models as MISOCPs . . . . . . 8

1.3 Application 2: Skewness in Portfolio Selection Models . . . . . . . 10

1.4 Application 3: Supply Chain Management . . . . . . . . . . . . . 11

1.5 Contributions to Literature . . . . . . . . . . . . . . . . . . . . . 13

1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . 13

2 Literature Review 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Mixed-Integer Second-Order Cone Programming . . . . . . . . . . 17

2.2.1 Second Order Cone Programming . . . . . . . . . . . . . . 17

2.2.2 Mixed-Integer Second-Order Cone Programming . . . . . . 24

2.2.3 MISOCPs Arising in Applications . . . . . . . . . . . . . . 34

2.3 Optimization Problems Arising in Portfolio Selection . . . . . . . 54

i

2.3.1 Markowitz Mean-Variance Framework . . . . . . . . . . . . 54

2.3.2 Single- and Multi-Period Portfolio Optimization . . . . . . 56

2.3.3 Revealed Preferences for Portfolio Selection . . . . . . . . 63

2.4 Optimization Problems Arising in Supply Chain Management . . 67

I Portfolio Optimization Models 72

3 Single- and Multi-Period Portfolio Optimization 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Single Period Portfolio Optimization Model . . . . . . . . . . . . 75

3.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . 76

3.2.2 Shortfall Risk Constraint . . . . . . . . . . . . . . . . . . . 78

3.2.3 Diversification By Sectors . . . . . . . . . . . . . . . . . . 80

3.2.4 Portfolio Constraints . . . . . . . . . . . . . . . . . . . . . 81

3.2.5 Buy-in-Threshold Constraints . . . . . . . . . . . . . . . . 83

3.2.6 Round-Lot Constraints . . . . . . . . . . . . . . . . . . . . 83

3.3 Multi-Period Portfolio Optimization Model . . . . . . . . . . . . . 84

3.3.1 Scenario Tree . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . 85

3.3.3 Transaction Costs Constraints . . . . . . . . . . . . . . . . 86

3.3.4 Shortfall Risk Constraints . . . . . . . . . . . . . . . . . . 87

ii

3.3.5 Diversification By Sectors . . . . . . . . . . . . . . . . . . 89

3.3.6 Portfolio Constraints . . . . . . . . . . . . . . . . . . . . . 90

3.3.7 Buy-in-Threshold Constraints . . . . . . . . . . . . . . . . 90

3.4 Solving the MISOCP . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.4.1 The Ratio Reformulation . . . . . . . . . . . . . . . . . . . 92

3.4.2 The Primal and Dual Penalty Problems . . . . . . . . . . 95

3.4.3 A Primal-Dual Penalty Interior-Point Method . . . . . . . 100

3.4.4 Warmstarting . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.4.5 Handling the discrete variables . . . . . . . . . . . . . . . 105

3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5.1 Numerical Results for the Single Period Model . . . . . . . 106

3.5.2 Numerical Results for the Multi-Period Model . . . . . . . 107

4 Revealed Preferences for Portfolio Selection Models 112

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2.1 Mean Variance Efficient Portfolios . . . . . . . . . . . . . . 114

4.2.2 Asset Allocation with Higher Moments . . . . . . . . . . . 116

4.3 Revealed Preferences . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

iii

II Models in Supply Chain Management 127

5 Optimization Problems Arising in Supply Chain Management 128

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Assumptions and notation . . . . . . . . . . . . . . . . . . . . . . 133

5.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.1 Shipment Policies . . . . . . . . . . . . . . . . . . . . . . . 137

5.3.2 TL policy model formulation . . . . . . . . . . . . . . . . . 138

5.3.3 LTL policy model formulation . . . . . . . . . . . . . . . . 141

5.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6 Conclusions and Future Research 152

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.1.1 Portfolio Selection Models as MISOCPs . . . . . . . . . . . 153

6.1.2 Skewness in Portfolio Selection Models . . . . . . . . . . . 154

6.1.3 Supply Chain Management . . . . . . . . . . . . . . . . . . 155

6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . 157

Bibliography 158

iv

List of Figures

1.1 Feasible Region of an MISOCP . . . . . . . . . . . . . . . . . . . 3

3.1 Scenario Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1 Distance to the Market Weights . . . . . . . . . . . . . . . . . . . 122

4.2 The Values of Risk Parameters . . . . . . . . . . . . . . . . . . . 124

5.1 Direct Shipping (LTL) vs. Peddling Shipping (TL) Policies . . . 137

5.2 Inventory-Time Plots for a Peddling Shipment Policy . . . . . . . 139

5.3 Inventory-Time Plots for a Direct Shipment Policy . . . . . . . . 142

5.4 Comparison of TL and LTL Shipment Policies . . . . . . . . . . . 148

5.5 Policy Comparison with 250% Increase in Setup Cost . . . . . . . 149

5.6 Policy Comparison with 500% Increase in Setup Cost . . . . . . . 150

5.7 Policy Comparison with 50% Reductions in TL Shipping Costs . 150

5.8 Policy Comparison with 100% Increase TL Shipping Costs . . . . 151

v

List of Tables

3.1 Results of the Branch-and-Bound Algorithm . . . . . . . . . . . . 108

3.2 Results of the Outer Approximation Algorithm for Single-Period . 108

3.3 Results of the Outer Approximation Algorithm for Multi-Period . 111

4.1 Distance to the Market Weights . . . . . . . . . . . . . . . . . . . 123

4.2 Risk Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.3 Current Dow Jones 30 Stocks . . . . . . . . . . . . . . . . . . . . 126

5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.2 Example Problem Parameters . . . . . . . . . . . . . . . . . . . . 144

5.3 Numerical Results for TL Shipment Policy . . . . . . . . . . . . . 146

5.4 Numerical Results for LTL Shipment Policy . . . . . . . . . . . . 147

vi

Acknowledgments

The research in this thesis resulted from collaborations with my advisor Pro-

fessor Hande Y. Benson and my committee members Professor Avijit Banerjee

and Professor Merrill Liechty. This thesis would not been possible without their

guidance, help and support.

I would like to express the deepest appreciation to Professor Hande Y. Benson,

who served as my academic and non-academic advisor during my entire time at

Drexel and provided constructive feedback for all my endeavors. She was always

there when I needed her. Without her guidance and persistent help I would not

be able to finish this thesis.

I would like to thank to my committee members Professor Oben Ceryan and

Professor Steven Weber for reading my thesis carefully and their valuable sugges-

tions and advice.

I would like to thank all my professors in Istanbul Bilgi University, Dr. Göksel

Asan, Dr. Orhan Erdem, Dr. Ipek Ö. Sanver, Dr. M. Remzi Sanver and Dr. Ege

Yazgan for inspiring me to the academic career.

I would like to especially thank my friends from Turkey, for their continu-

ous love, support, wishes and prayers. I would like to specifically acknowledge

Fatih Erçil, Mehmet Özkan, Yusuf Varlı, Fisun Cengiz, Said Salih Kaymakçı, and

vii

Neslihan Sakarya.

I would, next, like to thank my colleagues, Gülay Samatlı-Paç, Chang-Hyun

Kim, Ted Kim and Kurt Masten for making Ph.D. life more manageable. I also

would like to thank to all my friends; Alperen Ayhan, Süleyman Akocak, Selman

Erol, Fatih Karahan, Sercan Önal, Fazıl Paç and Mahmut Selman Sakar, for

making this county more enjoyable.

I would like to express my deepest gratitude to my fiancé Elif Ece Demiral

for giving me unlimited happiness and pleasure in the last two and a half years.

Thank you for being with me and for your all appreciated sacrifices.

I owe my deepest gratitude to my brother, Mehmet Saglam for his tremendous

support during my entire life. There are no words that express the depth of my

gratitude for everything he has ever done for me.

Finally, I am further indebted to my family, Nagihan Saglam, Yusuf Saglam,

Ilknur Saglam Altun, Ibrahim Altun, and Merve Sehiraltı Saglam for their in-

valuable support and unconditional love. I would like to specifically thank my

parents for their utmost dedication and unbounded sacrifice that helped me reach

this success. I want also to mention my nephew Kerem Altun who is such a great

source of joy in my life. I am always so grateful being a part of this great family.

This thesis is dedicated to my family.

viii

to my family

ix

CHAPTER 1. INTRODUCTION 1

Chapter 1

Introduction

This dissertation is on advanced mathematical programming with applications in

portfolio optimization and supply chain management. Specifically, we focus on

three types of problems arising as follows:

1. Portfolio optimization models with discrete decisions and risk constraints

modeled as cone constraints,

2. Inclusion of skewness in portfolio optimization frameworks as a Bayesian

decision problem, which can be modeled as a bilevel optimization problem.

3. Economic lot scheduling problem with discrete choices modeled as a mixed-

integer nonlinear programming problem (MINLP).

Within the scope of our study, we also observed a need for robust and efficient

methods for mixed-integer second-order cone programming problems (MISOCP)


to address application (1). Therefore, we conducted algorithmic development,

implementation and numerical studies to fill this gap. Since such methods for

bilevel optimization and MINLP already exist, along with efficient software, it

sufficed for us to be users rather than developers to address applications (2) and

(3).

1.1. Algorithmic Studies: MISOCP

In our algorithmic work, we study mixed-integer second-order cone programming

problems (MISOCPs) of the form

(1.1)minx∈X

cTx

s.t. ‖Aix+ bi‖ ≤ aT0ix+ b0i, i = 1, . . . , m

where x is the n-vector of decision variables, X = {(y, z) : y ∈ Zp, z ∈ Rk, p+k =

n}, and the data are c ∈ Rn, Ai ∈ Rmi×n, bi ∈ Rmi , a0i ∈ Rn, and b0i ∈ R for i

= 1,…,m. The notation ‖ · ‖ denotes the Euclidean norm, and the constraints are

said to define the second-order cone, also referred to as the Lorentz cone.

MISOCP is a category of mixed-integer nonlinear optimization (MINLP) prob-

lems, where there is a special structure. For this category of problems, we want to

minimize a linear objective function, subject to second-order constraint(s). Note

that this form also accommodates, and generally includes, linear constraints: if

mi = 0, the ith constraint is linear. The decision variables can be continuous or


Figure 1.1: Feasible Region of an MISOCP are horizontal slices of the Lorentz

Cone

discrete, and while the form of 1.1 allows cases where p = 0 or k = 0, we are only

interested in those cases where p > 0. The feasible region of an MISOCP with

p = 1, k = 2 consists of the slices of the ice-cream cone, as shown in Figure 1.1.

When p = 0, 1.1 reduces to the continuous problem referred to a second-order

cone programming problem (SOCP). SOCPs have been well studied in literature,

and computationally efficient implementations of solution algorithms exist. We

will provide a thorough survey of these algorithms in Section 3.4 as characteris-

tics of these algorithms will greatly impact the methodologies developed for this

dissertation. Of relatively new interest to the research community is the exten-

sion to MISOCP fueled by interest in portfolio optimization problems in business


and network reliability models in engineering. Comparatively, MISOCP is a less

mature field than SOCP, and this dissertation is among the first ones to branch

into this exiting area.

Although this field is only 5-6 years old, this class of problems arise in a vari-

ety of important application areas ranging from finance to electrical engineering,

from operations management to statistics. In Section 2.2.3, we will discuss these

important applications areas in detail, but we will list several examples here as

well to show their importance:

• In [121], Pinar studies a multiperiod pricing problem for an American option

under uncertainty where the objective function is to maximize at the end

of period expected wealth subject to the second-order cone constraints that

arise as risk constraints providing a lower bound for the Sharpe ratio of

the final wealth position of the buyer. The binary variables are introduced

to denote the decision whether to exercise the option at each node of the

scenario tree, and additional constraints enforce that the option is exercised

at no more than 1 node in each sample path. (Please see Section 2.2.3.1 for

more detail about the application.)

• In [42], Cheng et.al. present multi-point transmission problem for cellular

networks. The binary variables represent the assignment of mobile units to

base stations, where multiple base stations can coordinate the transmission.


The second-order cone constraints are formulated for each base station and

serve to limit the total power transmitted from the base station to all of the

mobile units that it serves. (Please see Section 2.2.3.4 for more detail about

the application.)

• In [132], Taylor and Hover consider several problems from power distribu-

tion system reconfiguration. The second-order cone constraints arise in the

approximation of flow distribution equations. The binary variables appear

as switching variables. (Please see Section 2.2.3.5 for more detail about the

application.)

• In [31], Brandenberg and Roth propose a new algorithm for the Euclidean

k-center problem. The binary variables denote the assignment of the points

to the balls, and second-order cone constraints are used to denote that if a

point is assigned to a ball, then the Euclidean distance between the point

and the center of the ball must be no more than the radius of the ball.

(Please see Section 2.2.3.7 for more detail about the application.)

• In [55], Du et.al. present an MISOCP as a relaxation of the MINLP that

arises in the problem of determining the berthing positions and order for

a group of vessels waiting at a container terminal in order to minimize the

total waiting time of the vessels. Binary variables are used to denote the

relative positions of pairs of vessels (whether one vessel is to the left of


another and whether one vessel is earlier than another.) The second-order

cone constraints arise in a reformulation of a nonlinear fuel consumption

constraint. (Please see Section 2.2.3.9 for more detail about the application.)

As shown above, MISOCPs are very common in a variety of application areas,

because of two reasons: (1) risk (volatility) constraints can easily be formulated

as second-order cone constraints, and (2) binary choices and discrete decisions

are quiet common in the real world. Then the question arises: Despite the many

important application areas of this class of problems, why has there been so little

development this field until the last decade? There are several natural reasons for

this:

• About two decades ago, SOCPs became very popular, as this class of prob-

lems arises in important application areas ranging from financial engineering

to electrical engineering, as well. However due to their special structure,

SOCPs were viewed as extensions of linear programming problems over the

Lorenz cone, instead of as nonlinear programming problems. While this

feature was essential in the development of algorithms for SOCP, the non-

linear nature of the cone constraints complicates the successful extension

of similar concepts from mixed-integer linear programming, such as column

generation and cutting plane methods, to MISOCP. Moreover, viewing MIS-

OCPs as mixed-integer nonlinear problems (MINLP) has had its share of


challenges as well, due to MINLP algorithms requiring twice continuously

differentiable constraint functions, a property violated by the constraints of

1.1. On a more basic level, due to their long treatment as LP extensions,

SOCPs were largely unknown by the NLP community, and by extension,

MISOCPs were largely unknown by the MINLP community.

• Compared to two decades ago, today we have more powerful computers that

allow us to solve large-scale complex optimization problems. Here, the pro-

pose a branch-and-bound algorithm to handle the discrete variables in the

single-portfolio optimization problem, which is formulated as an MISOCP.

We may need to solve up to 2n − 1 subproblems, where n is the number of

discrete variables in the model, for this algorithm. Due to the previously dis-

cussed application areas leading to large-scale problems, the computational

studies conducted for this dissertation may not have been possible before.

In the MISOCP framework, we need to handle two different types of con-

straints, second-order cone and discrete constraints that bring extra difficulty to

the problem. In this dissertation, we use interior-point methods, which require

twice continuous differentiability,so we propose the ratio reformulation to rewrite

the cone constraint to obtain a smooth convex formulation, for solving portfo-

lio optimization problems. We propose two MINLP methods, branch-and-bound

and outer approximation to handle discrete variables. The primal-dual penalty


method is applied to the interior-point algorithm to enable warmstarts and infea-

sibility detection. We investigate the application of our proposed techniques to

portfolio optimization problems that can be formulated as MISOCPs.

1.2. Application 1: Portfolio Selection Models as

MISOCPs

Classical portfolio selection models are based on the Markowitz mean/variance

framework (1.2), where there is a trade-off between expected return and the risk

that the investor may be willing to take on, in a single-period time horizon.

(1.2)

maxw

rTw

s.t. wTQw ≤ σ2

∑ni=1wi = 1

g(w) ≤ 0

w ≥ 0

Although there have been substantial developments in portfolio selection models

since the publication of [103], there is a huge gap between the theoretical work

and real world application. Therefore, there is still work to be done to incorpo-

rate more complex components into these models and solve them efficiently and

reliably. In this dissertation, our aim is to model more realistic scenarios when an

investor experience when he is constructing his portfolio. Therefore, we have cho-


sen to incorporate transaction costs, conditional value-at-risk (CVaR) constraints,

diversification requirements by sectors, and buy-in-thresholds into our framework.

These model components/features have been adapted from [1], [29], [67], [73], and

[99], and we come up with a very comprehensive portfolio selection model in the

portfolio optimization literature. The first two components of the model require

the use of second-order cone constraints while the latter two are implemented using

binary variables, resulting in an MISOCP. In Chapter 3, Section 3.2 we attempt to

solve these comprehensive problems. We propose two algorithms for MISOCP: one

with a branch-and-bound framework and the other with an outer approximation

framework, both using a primal-dual penalty interior-point method to solve the

underlying SOCPs. Both algorithms can accommodate the various cuts appearing

in MISOCP literature for further improvement, and they take into account issues

such as the non-differentiability of the underlying SOCP, warm-starting, and in-

feasibility detection. We have implemented both branch-and-bound and outer

approximation frameworks that use this method, and use them to solve single-

period portfolio optimization problems that can be formulated as MISOCPs. In

addition, in Section 3.3 we further extend this model to the multi-period case that

is obtained using a binary scenario tree that is constructed with monthly returns

of the closing price of the stocks from the S&P 500. We solve these models with

the MATLAB-based Mixed Integer Linear and Nonlinear Optimizer (MILANO)

solver that implements a variety of methods for handling integer variables, cone


constraints, linear and nonlinear subproblems. The real-world data are used for

the numerical examples. Numerical results show that we can solve instances with

up to 400 stocks successfully. The infeasibility detection capability provided by

the primal-dual penalty approach allows us to either solve or declare infeasibility

at each node, thereby leading to a robust method. The warm-start capability is

shown to significantly improve algorithm efficiency.

1.3. Application 2: Skewness in Portfolio Selection

Models

In Chapter 4, we consider the paradigm of mean/variance efficient portfolios where

the investor’s objective function is to choose the portfolio weights to maximize ex-

pected return subject to predetermined level of risk. Posing the portfolio selection

problem as a Bayesian decision problem we investigate which reasonable assump-

tions on agent’s utility and the underlying probability model lead to asset alloca-

tion rules that depend on mean, variance and skewness of future returns. The main

contribution of this article is the empirical validation of this argument by compar-

ing it with traditional mean/variance efficient portfolios. In this study, we consider

two competing descriptions of portfolio selection, the traditional mean/variance

efficient portfolio versus a generalization allowing for decision makers to consider


skewness in their asset allocation. We develop a framework to attempt explaining

observed investor preferences by the two alternative utility functions. Minimizing

the discrepancy between the optimal decision under the considered utility func-

tions and the observed data formalizes the comparison. The discrepancy between

the market weight and optimal portfolio weight is formulated as an SOCP and the

model becomes bilevel second-order cone programming (BLSOCP) problem where

the constraint of an upper level optimization problem, is also an optimization prob-

lem. In our framework, the outer problem is to maximize the investor’s objective

function which is penalized by risk for the first model and it is constrained by both

variance/covariance and skewness matrices for the second model. For the inner

problem, we solve an optimization problem, minimizing the discrepancy between

market and the optimal portfolio. The described algorithm is highly computa-

tion intensive. Our numerical experiments are conducted on portfolio drawn from

30 different stocks from the Dow Jones. Numerical results show that investor’s

preferences are better explained when skewness is taken into account.

1.4. Application 3: Supply Chain Management

The advanced optimization modeling techniques and algorithms we used for port-

folio optimization extended naturally to cover some models in supply chain man-

agement. Today’s supply chains are impacted by increased complexity, unpre-


dictable economic conditions, operational risks, environmental regulations, glob-

alization and rising fuel costs. Historically, optimization projects within the sup-

ply chain have been cumbersome, time-consuming undertakings. Many companies

find themselves in a constant struggle to maintain efficiency at every stage along

the supply chain, attempting to reduce costs and increase productivity within their

procurement-production-distribution networks, in the face of intense competitive

pressures. In this context, holistic integration of decisions involving serial stages

of activities has received attention from researchers in recent years. We attempt

to extend the classical and well-known economic lot scheduling problem (ELSP)

by incorporating the transportation decision, accounting for finished goods inven-

tories in discrete, sizeable lots. In Chapter 5 we formulate mathematical models

that attempt to integrate the production lot scheduling problem with outbound

shipment decisions. The optimization objective is to minimize the total relevant

costs of a manufacturer, which distributes a set of products to multiple retailers.

In making the production/distribution decisions, the common cycle approach is

employed to solve the ELSP, for simplicity. Two different shipping scenarios, i.e.

periodic full truckload (TL) peddling shipments and less than truckload (LTL)

direct shipping, are integrated with and linked to the multi-product batching de-

cisions. The resulting mixed-integer, non-linear programming models (MINLPs)

are solved by the BONMIN solver. We illustrate and evaluate a set of numerical

examples to find the relative efficiencies of these policies.


1.5. Contributions to Literature

In summary, this thesis provides:

• a thorough survey of the literature on applications of and algorithms for

MISOCP

• proposed branch-and-bound and outer approximation algorithms to handle

discrete variables in this class of problems

• comprehensive portfolio selection models which have included transaction

cost with risk constraint and more realistic diversification requirements

• better explanations of investors’ preferences when skewness is taken into

account for the portfolio selection models

• an insight to supply chain practitioners about the importance of integrat-

ing the production schedule with transportation planning and selecting an

appropriate method of distribution.

1.6. Organization of the Thesis

After providing a survey of all related literature, we have divided the dissertation

into two parts. The first part covers the portfolio selection model as a MISOCP

and all the accompanying algorithmic and computational studied for this class of


problems. To continue with the theme of portfolio optimization, we also include

our work evaluation of the inclusion of skewness in these models. The second part

covers models from supply chain management.

As such, the balance of this thesis is organized as follows:

• Chapter 2 provides a thorough literature review of three different streams of

existing literature: mixed-integer second-order cone programming (2.2), op-

timization methods arising in portfolio optimization (2.3) and supply chain

management (2.4), that provide an overall view of the concepts that will be

utilized in the subsequent chapters of this dissertation.

• As mentioned, there are two chapters in the first part of the dissertation. In

Chapter 3, we consider single- and multiperiod portfolio optimization with

second-order cone constraint and discrete decisions. In Chapter 4, we study

the effect of skewness of the future returns for the portfolio selection models.

• In Part II, Chapter 5, we formulate mathematical models that attempt to

integrate the production scheduling problem with the outbound decision.

• Finally Chapter 6 provides summary and some concluding remarks for each

application as well as some future research directions regarding to the algo-

rithmic work that we will present.

CHAPTER 2. LITERATURE REVIEW 15

Chapter 2

Literature Review

2.1. Introduction

This chapter first presents a review of three different streams of the existing lit-

erature: mixed-integer second-order cone programming (2.2), and optimization

methods arising in portfolio optimization (2.3) and supply chain management

(2.4), that provide an overall view of the concepts that will be utilized in the

subsequent chapters of this dissertation.

In the next section, we provide a brief overview of relevant solution methods

for SOCP, since the choice of method for the underlying continuous relaxation

greatly influences the design and performance of the overall solution method for

MISOCP. These solution methods include interior-point methods designed specif-

ically for SOCP, adaptations of interior-point methods for nonlinear programming


to the case of SOCP, and lifted polyhedral relaxations, and they are incorporated

into the MISOCP algorithms presented in Section 2.2.2. In general, MISOCP

algorithms fall into two groups: extensions of MILP approaches (since the second-

order cone can be viewed as an extension of the linear cone) or special-purpose

MINLP approaches (since SOCPs can also be viewed as nonlinear programming

problems with special structure). As such, we will present an overview of cuts,

including extensions of Gomory and rounding cuts, and relaxations for MISOCP,

while discussing the adaptation of branch-and-cut, branch-and-bound, and outer

approximation methods to the case of this class of problems. In Section 2.2.3, we

start the literature review on applications ranging from scheduling to electrical en-

gineering, from finance to operations management. Several of the examples arise

as reformulations or even relaxations of MINLPs, as MISOCPs can sometimes

present advantages over MILP in this regard.

In Section 2.3, we start the literature review on portfolio optimization. In

Chapters 3 and 4, we consider a single-period portfolio optimization problem

which is based on the Markowitz mean-variance framework [104], where there is

a trade-off between expected return and risk (market volatility) that the investor

may be willing to take on. Therefore, we provide thorough literature review

on mean-variance framework in 2.3.1. In Sections 2.3.2 and 2.3.3, we provide

the additional literature directly related to each of our models in Chapter 3 and

Chapter 4, respectively.


In Section 2.4, we provide a review of the relevant research literature on multi-

product batch production and truck shipment scheduling under different shipping

policies, which is presented in Chapter 5 in detail.

2.2. Mixed-Integer Second-Order Cone Programming

In this section, we provide a thorough overview of the existing literature on MIS-

OCPs. While the field may not be as mature as SOCP, there have been a wide

variety of algorithms studied in the last decade, and the application areas range

from supply chain management to electrical engineering to asset pricing. we will

focus on portfolio optimization models in Chapter 3 and will provide a literature

review for this particular application area as well.

2.2.1. Algorithms for Second Order Cone Programming

In this section, we give a brief overview of several algorithms for solving the

underlying SOCPs. These algorithms will have a significant impact on the design

and efficiency of the overall MISOCP methods that will be discussed in the next

section. For a thorough overview of SOCP, including theory, applications, and

solution algorithms, we refer the reader to [2].


2.2.1.1. Interior-Point Methods for SOCP

The continuous relaxation of (1.1) is given by a problem of the same form as

(1.1), but with x ∈ Rn. In order to write the dual problem, let us first introduce

auxiliary variables (t0i, ti) ∈ Rmi+1 for i = 1, . . . , m and rewrite the continuous

relaxation of (1.1) as

(2.1)

minx,t0,t

cTx

s.t. t0i = aT0ix+ b0i, i = 1, . . . , m

ti = Aix+ bi, i = 1, . . . , m

‖ti‖ ≤ t0i, i = 1, . . . , m.

The dual problem can now be written as

(2.2)

maxλ0,λ

m∑

i=1

(bTi λi + bT0iλ0i)

s.t.m∑

i=1

(ATi λi + a0iλ0i) = c

‖λi‖ ≤ λ0i, i = 1, . . . , m,

where (λ0i, λi) ∈ Rmi+1 for i = 1, . . . , m are the dual variables. Assuming that we

have strict interiors for (2.1) and (2.2), strong duality holds and the optimality


conditions for (2.1) are

(2.3)

t0i = aT0ix+ b0i, i = 1, . . . , m

ti = Aix+ bi, i = 1, . . . , mm∑

i=1


‖ti‖ ≤ t0i, i = 1, . . . , m

‖λi‖ ≤ λ0i, i = 1, . . . , m

(t0i, ti) ◦ (λ0i, λi) = 0, i = 1, . . . , m,

where

(t0i, ti) ◦ (λ0i, λi) = (tTi λi, t0iλi + λ0iti)T .

As with linear programming, an interior-point method starts by introducing a

barrier parameter µ > 0, perturbing the last (complementarity) condition in (2.3)

as

(t0i, ti) ◦ (λ0i, λi) = 2µei, i = 1, . . . , m,

with ei =

1

0mi

, and initializing t and λ on the strict interior of the second-order

cone. The Newton system associated with the perturbed conditions

t0i = aT0ix+ b0i, i = 1, . . . , m

ti = Aix+ bi, i = 1, . . . , mm∑

i=1


(t0i, ti) ◦ (λ0i, λi) = 2µei, i = 1, . . . , m,


is solved at each iteration, but a scaling, such as HRVW/KSH/M ([82],[88],[110]),

AHO [3], or NT ([114],[115]), may be needed to do obtain iterates on the interior

of the second-order cone. The barrier parameter is also reduced at each iteration.

Optimality is declared when (2.3) are satisfied to within a small tolerance.

Interior-point methods for SOCP are theoretically robust and computationally

efficient. However, there are drawbacks when they are used within an MISOCP

framework, including the need to start from a strictly feasible primal-dual pair

of solutions and the accuracy level of the optimal solution obtained. The former

makes it hard to warmstart the algorithm from a previously obtained solution,

while the latter may create issues with declaring feasibility with respect to the

integer variables and adding cuts to the underlying SOCP.

2.2.1.2. Extensions of Interior-Point Methods for NLP to SOCP

While SOCP can be seen as an extension of linear programming, it can also be

seen as a special case of nonlinear programming (NLP). The formulation (2.1) is

already in the form of a structured, convex NLP. Therefore, another possibility for

a solution algorithm is to use interior-point methods that have been developed for

NLP. However, an interior-point method for NLP requires that all objective and

constraint functions be twice continuously differentiable, so the main challenge in

using such a method is the nondifferentiability of the second-order cone constraint

functions due to the use of the Euclidean norm.


In [23], Benson and Vanderbei investigated the nondifferentiability of an SOCP

and proposed several reformulations of the second-order cone constraint to over-

come this issue. Note that the nondifferentiability is only an issue if it occurs at

the optimal solution. Since an initial solution can be randomized, especially when

using an infeasible interior-point method to solve the SOCP, the probability of

encountering a point of nondifferentiability is 0.

For a constraint of the form

(2.4) ‖ti‖ ≤ t0i,

Benson and Vanderbei proposed the following:

• Exponential reformulation: Replacing (3.4) with e(tTiti−t2

0i)/2 ≤ 1 and t0i ≥ 0

gives a smooth and convex reformulation of the problem, but numerical

issues frequently arise due to the exponential.

• Smoothing by perturbation: Introducing a scalar variable v into the norm

gives a constraint of the form√v2 + tTi ti ≤ t0i, but in order for the formu-

lation to be smooth, we need v > 0. This is ensured by setting v ≥ ǫ for a

small constant ǫ, usually taken around 10−6 − 10−4.

• Ratio reformulation: Replacing (3.4) with tTi tit0i

≤ t0i and t0i ≥ 0 yields a

convex reformulation of the problem, but the constraint function may still

not be smooth. Nevertheless, in many applications, such as the portfolio


optimization problems to be studied in the next section, the right-hand side

of the second-order cone constraint in (1.1) is either a scalar or bounded

away from zero at the optimal solution.

Once the reformulation is complete, the problem can be solved using any

variant of an interior-point method. In [23], Benson and Vanderbei used the

infeasible primal-dual interior-point method that was implemented in loqo [135].

Introducing a barrier parameter µ > 0 and slack variables w, s ≥ 0, the perturbed

optimality conditions for an SOCP that has undergone the ratio reformulation

can be expressed as

(2.5)

t0i = aT0ix+ b0i, i = 1, . . . , m

ti = Aix+ bi, i = 1, . . . , mm∑

i=1


tTi tit0i

+ w = t0i, i = 1, . . . , m

λTi λiλ0i

+ s = λ0i, i = 1, . . . , m

wisi = µ, i = 1, . . . , m.

Starting at an initial solution with w, s > 0, we solve the Newton system associated

with (2.5) at each iteration, find an appropriate steplength using a merit function

or a filter, and update the barrier parameter as needed. The algorithm stops when

it satisfies (2.5), with µ sufficiently close to 0, to a desired level of accuracy.

Using this approach, the accuracy of the solution to the SOCP still remains a


concern. However, due to recent work in the area, the warmstart issue is starting

to get resolved. I refer the reader to [15] for details on a primal-dual penalty

method that enables warmstarts when solving SOCPs.

2.2.1.3. Lifted Polyhedral Relaxation

A very different perspective on (approximately) solving SOCPs is to employ a

polyhedral relaxation of the convex feasible region and to solve a related linear

programming problem instead. However, in doing so, it is important to ensure

that the size of the linear programming problem remains tractable. Ben-Tal and

Nemirovski [12] have presented a lifted polyhedral relaxation that uses a polyno-

mial number of constraints and auxiliary variables, and this relaxation method

has been further refined by Glineur [70].

Starting with an SOCP of the form (2.1), let us focus on the constraint ‖ti‖ ≤

t0i for some i. The goal is to construct a polyhedron that is ǫ-tight, i.e. satisfies

‖ti‖ ≤ (1 + ǫ)t0i for a small ǫ > 0. For ease of presentation, we assume that mi is

an integer power of 2. (We refer the reader to the details provided in [12] and [70]

for the case when mi is not an integer power of 2.) If the variables are grouped

into r/2 pairs and an auxiliary variable ρj is associated with the jth pair, then

the set of points satisfying the original cone constraint canbe rewritten as

{(t0i, ti) ∈ Rmi+1 : ∃ρ ∈ Rmi/2 s.t. ρTρ ≤ t20i, t2i(2j−1) + t2i(2j) ≤ ρ2

j , j = 1, . . . , mi}.


This new definition uses one cone of dimension mi/2 + 1 and mi/2 cones of di-

mension 3. This process is recursively applied to cone of dimension mi/2 + 1,

until there are only 3-dimensional cones left. Then, each 3-dimensional cone can

be replaced with a polyhedral relaxation of the form

(2.6) {(r0, r1, r2) ∈ R3 : r0 ≥ 0 and ∃(α, β) ∈ R2s s.t.

r0 = αs cos(π2s

)+ βs sin

(π2s

)

α1 = r1 cos(π) + r2 sin(π)

β1 ≥ |r2 cos(π) − r1 sin(π)|

αi+1 = αi cos(π2i

)+ βi sin

(π2i

), i = 1, . . . , s− 1

βi+1 ≥∣∣∣βi cos

(π2i

)− αi sin

(π2i

)∣∣∣ , i = 1, . . . , s− 1,

for some s ∈ Z.

Given that the resulting problem is a linear program, it can be solved using

any number of suitable methods, including the simplex method or a crossover

approach, both of which would yield very efficient warmstarts.

2.2.2. Algorithms for Mixed-Integer Second-Order Cone Pro-

gramming

One very straightforward way to devise a method for solving MISOCPs is to use a

branch-and-bound algorithm that calls an interior-point method designed specifi-

cally for SOCPs at each node. However, if such a method is to be competitive on


large-scale MISOCPs, it is important to reduce the number of nodes in the tree

using cuts and relaxations designed specifically for MISOCP and to reduce the

runtime at each node using an SOCP solver that is capable of warmstarting and

infeasibility detection.

There are other approaches for MINLP besides branch-and-bound which can

similarly be adopted for the case of MISOCP, using the fact that the underlying

SOCPs are essentially convex NLPs. These approaches include outer approxi-

mation [57], extended cutting-plane methods [139], and generalized Benders de-

composition [69]. However, the nondifferentiability of the constraint functions in

(1.1) is of particular concern when generating the gradient-based cuts required by

these methods, and their application to MISOCP should be done carefully and by

considering this special case.

Additionally, any method that can convert the underlying SOCPs into linear

programming problems can take advantage of the efficient algorithms designed for

MILP.

A number of studies appear in literature dealing with algorithms specifically

for MISOCP, and we will now present them here.

2.2.2.1. Gomory Cuts and Tight Relaxations

In [39], Cezik and Iyengar study mixed-integer conic programming problems (MICPs),

of which both mixed-integer linear programming problems and MISOCPs are sub-


sets. Their approach is to extend some well-known techniques for mixed-integer

linear programming to mixed-integer programs involving second-order cone and/or

semidefinite constraints. Since the problem setup in [39] includes a more general

cone than ours, we have adapted their discussion to the case of the second-order

cone.

Their first extension is that of Gomory cuts to integer conic programs. For

the case of integer SOCPs, they note that

(2.7){x ∈ Rn : ‖Aix+ bi‖ ≤ aT0ix+ b0i, i = 1, . . . , m

}⇔

x ∈ Rn :

m∑

i=1

(a0iu0i +n∑

j=1

aTijui)

T

x ≥m∑

i=1

(b0iu0i + bTi ui), (u0i, ui)T ∈ K∗

i , i = 1, . . . , m

,

where Ai = [ai1, ai2, . . . , ain] and K∗i is the dual cone of the ith second order cone.

This equivalence leads to the following natural extension of the Chvatal-Gomory

procedure for integer SOCPs:

1. Choose (u0i, ui)T ∈ K∗i , i = 1, . . . , m. Then,

m∑

i=1

(a0iu0i +n∑

j=1

aTijui)

T

x ≥m∑

i=1

(b0iu0i + bTi ui).

2. Without loss of generality, x ≥ 0, so

m∑

i=1

(⌈a0i⌉u0i +n∑

j=1

⌈aij⌉Tui)T

x ≥m∑

i=1

(b0iu0i + bTi ui).

3. By the integrality of x, it holds that

m∑

i=1

(⌈a0i⌉u0i +n∑

j=1

⌈aij⌉Tui)T

x ≥m∑

i=1

(⌈b0i⌉u0i + ⌈bi⌉Tui)


is a valid linear inequality that can be added to the cone constraints.

The authors also prove that every valid inequality for the convex hull of the feasible

region of an integer SOCP can be obtained by repeating the above procedure a

finite number of times.

The second extension is that of sequential convexification to the case of integer

conic programs. This approach, which was studied in [7], [126], [127], [100], [92],

for pure and mixed-integer linear programming problems can provide tighter relax-

ations than the continuous relaxation of the integer SOCP. To extend the Lovasz-

Schrijver and Balas-Ceria-Cornuejols hierarchies, the authors start by picking a

subset of size l of the variables and introduce Y 0 = [y01 . . . y

0l ] and Y 1 = [y1

1 . . . y1l ]

with

y0k = (1 − xjk

)

1

x

, y1

k = xjk

1

x

, k = 1, . . . , l.


Then, the following is a relaxation for (1.1) with all binary variables:

min cTx

s.t. y0k + y1

k =

1

x

, k = 1, . . . , l

‖n∑

j=1

y0jkaij + y0

0kbi‖ ≤n∑

j=1

y0jka0j + y0

0kb0, k = 1, . . . , l

‖n∑

j=1

y1jkaij + y1

0kbi‖ ≤n∑

j=1

y1jka0j + y1

0kb0, k = 1, . . . , l

y0kk = 0, k = 1, . . . , l

y1kk = y1

0k, k = 1, . . . , l

Y 1 = (Y 1)T .

To extend the Sherali-Adams and Laserre hierarchies, the authors also start

by picking a subset of size l of the variables and call this subset B. Let y be a

vector that is indexed by the empty set, subsets H ⊆ B, and sets of the form

H ∪ {j} for j not picked for B, and define y as follows:

yI =

1, I = ∅∏

j∈I

xj , otherwise.

Then, define zI0 ∈ R and zI ∈ Rn for I ⊆ B:

zI0 =∏

j∈I

xj∏

j∈B\I

(1 − xj) =∑

I⊆H⊆B

(−1)|B\H|yH ≥ 0,

zIk = xk∏

j∈I

xj∏

j∈B\I

(1 − xj) =∑

I⊆H⊆B

(−1)|B\H|yH∪{k}, k = 1, . . . , n.


Thus, the following problem is a relaxation of the binary SOCP:

min cTx

s.t. xj = y{j}, k = 1, . . . , n

‖AizI + bizI0‖ ≤ aT0iz

I + b0izI0 , I ⊆ B.

Additional hierarchies based on these principles are also discussed in the paper.

The authors propose a cut algorithm can use the Chvatal-Gomory procedure

and the tighter relaxations. However, the success of this algorithm is rather limited

since the authors consider only interior-point methods for SOCP as the solution

algorithm for the underlying SOCPs and implement it using SeDuMi. As they

note, the use of interior-point methods results in a solution that is feasible sub-

ject to a tolerance and may need rounding prior to applying the cut generation

procedure. In addition, warmstarts from feasible dual solutions are not available

within SeDuMi, as is the case for most other codes for mixed-integer conic pro-

grams. Noting these limitations, the authors present preliminary numerical results

and pointers for future improvement.

2.2.2.2. Rounding Cuts

In [5], Atamturk and Narayanan focus on MISOCPs and their solution using a

branch-and-bound framework. They introduce rounding cuts obtained by first

decomposing each second-order cone constraint into polyhedral sets. In order to

introduce this approach, we first note that according to the definition of (1.1), the


variable x can be decomposed into (y, z) : y ∈ Zp, z ∈ Rk, p + k = n. For each

second-order cone i = 1, . . . , m, and partitioning the columns of Ai into Ayi and

Azi and the vector a0i into ay0i and az0i, the constraints of (2.1) can be rewritten as

follows:

t0i ≤ (ay0i)Ty + (az0i)

T z + b0i

ti ≥ |Ayi y + Azi z − bi|

‖ti‖ ≤ t0i.

We assume that the absolute value in the second constraint is elementwise and

focus on one such constraint which we will write as

(2.8) |ayy + azz + b| ≤ t

where ay is a row of Ayi for some i, az is the corresponding row in Azi , and b and

t are the corresponding elements of bi and ti, respectively. This form is both for

ease of notation and to better match the exposition in [5]. The set S is defined

as {y ∈ Zp, z ∈ Rk, t ∈ R : |ayy + azz + b| ≤ t, y ≥ 0, z ≥ 0}, where the

nonnegativities of y and z can be imposed without loss of generality (i.e., if a

variable is free, we can always split it into two nonnegative ones). Grouping the

terms of azz with positive and negative coefficients into z+ and z−, respectively,

(2.8) is rewritten as

(2.9) |ayy + z+ − z− + b| ≤ t.


The authors first define a rounding function ϕf for 0 ≤ f < 1 as

ϕf(v) =

(1 − 2f)n− (v − n), if n ≤ v < n+ f

(1 − 2f)n+ (v − n) − 2f, n + f ≤ v < n + 1,

where n ∈ Z. Then, they show that the following is a valid inequality for S

n∑

j=1

ϕf(aj/α)yj − ϕf (b/α) ≤ (t+ z+ + z−)/|α|

for any α 6= 0 and f = b/α−⌊b/α⌋. In addition, if b/ai > 0 for some i and α = ai,

then the above inequality is shown to be facet-defining for the convex hull of S.

These rounding cuts are added at the root node of the branch-and-bound tree,

and the preliminary results in [5] show that the cuts can significantly reduce the

number of nodes in the tree. The authors provide a more thorough analysis and

further examples showing the success of their approach in [6].

2.2.2.3. MILP methods applied to lifted polyhedral relaxation

In [137], Vielma et.al. propose using a lifted polyhedral relaxation ([12], [70], and

described in Section 3.3) of the underlying SOCPs, thereby solving the MISOCPs

using a linear programming based branch-and-bound framework. Their approach

can be generalized to any convex MINLP, does not use gradients to generate the

cuts, and benefits from the linear programming structure that can use a simplex-

based method with warmstarting capabilities within the solution process.

The Lifted LP Branch-and-Bound Algorithm presented in [137] is too detailed

to present in its entirety here, so we will give a brief outline and the interested


reader is referred to [137], particularly Figure 1 of that paper. In general, the al-

gorithm proceeds as the usual branch-and-bound method for MILPs by branching

on discrete variables with non-integer values, except solving the lifted polyhedral

relaxation of the associated SOCP at each node. If a feasible solution is found at

any node, the continuous relaxation of the MISOCP is solved at that node to see

if the exact solution (rather than an ǫ-tight relaxation), still yields a feasible solu-

tion. If so, the node is fathomed by integrality. Otherwise, we continue branching

on a discrete variable with a non-integer value. This approach ensures that only

linear programming problems are solved at most nodes of the tree and limits the

solution of the underlying SOCPs to a much smaller number of nodes.

Numerical studies on portfolio optimization problems show that the method

outperforms CPLEX and Bonmin. A similar method is used in [128] by Soberanis

to solve the MISOCP reformulations of risk optimization problems with p-order

conic constraints.

2.2.2.4. Extensions of Convex MINLP methods to MISOCP

In [54], Drewes proposes both a branch-and-cut method and a hybrid branch-and-

bound/outer approximation method for solving MISOCPs. The branch-and-cut

method uses techniques similar to [39] and those developed in [130] for mixed-

integer convex optimization problems with binary variables. Therefore, given

its similarity to the method presented in Section 4.1, we will not present this


approach, but instead provide details on the hybrid branch-and-bound/outer ap-

proximation method. Numerical results for both methods are provided in [54] for

a number of test problems.

The hybrid approach extends outer approximation methods, which use gradient-

based techniques to generate cuts, to the case of MISOCPs using subgradients.

As in outer approximation, constraints of the form ‖ti‖ ≤ t0i are replaced by

(‖ti‖ − t0i‖) + ξTi (ti − ti) + ξ0i(t0i − t0i) ≤ 0,

where (t0i, ti) is part of the solution of a continuous relaxation of the MISOCP

and (ξ0, ξ) is a subgradient of the second-order cone constraint function ‖ti‖ − t0i

at (t0i, ti). If ti 6= 0, the gradient can be used and set

ξ0i = −1 and ξi =ti

‖ti‖.

Otherwise, the dual variables (λ0i, λi) can be used to get an appropriate subgra-

dient. In [54], Drewes proposes that

ξ0i = −1 and ξi =

− λi

λ0i, if λ0i > 0

0, otherwise.

Additional cuts are generated from infeasible instances and are described in [54].

In [14] and [15], Benson and Saglam propose two MINLP methods, branch-

and-bound and outer approximation, for solving MISOCPs. Since the underlying

problems are smoothed using the ratio reformulation, as described in Section


3.2, and a primal-dual penalty method is applied to the interior-point algorithm

to enable warmstarts and infeasibility detection, both MINLP methods can be

applied directly and efficiently to solve an MISOCP. Preliminary numerical results

on problems arising in portfolio optimization are encouraging.

2.2.3. MISOCPs Arising in Applications

In this section, we give an overview of MISOCPs arising in a variety of application

areas in business, engineering, and statistics. It should be noted that this is only a

representative list and not an exhaustive one by any means. One of the challenges

in gathering a literature review on MISOCPs is that, many times, authors do not

recognize the special structure of the problem and simply identify the model as a

MINLP, solved using traditional MINLP methods. Therefore, we have included

in this section only those models that have been recognized by the authors as

MISOCPs.

We should also note that due to the wide variety of models, each subsection

below will have a self-contained list of notation. We will return to the formulation

(1.1) and related notation in the next section.

2.2.3.1. Options Pricing

In [121], Pinar describes a pricing problem for an American option in a financial

market under uncertainty. The multiperiod, discrete time, and discrete state space


structure is modeled using a scenario tree, and, therefore, the resulting problem

is large-scale. The set of nodes is denoted by N , and the nodes corresponding to

time period t are denoted by Nt. The planning horizon is at time T . π(n) denotes

the parent node of n, and A(n) denotes the ascendant nodes of n, including itself.

The probability of each n ∈ NT is denoted by pn.

We assume that there is a market consisting of J + 1 securities, with prices

at node n given by zn = (z0n, z

1n, . . . , z

Jn)T . The security with index 0 is assumed

to be risk-free. The decision variables θn ∈ RJ+1 denote the portfolio allocations

at node n, and thus, zTn θn denotes the value of the portfolio at the node. The

binary decision variables en indicate whether an American option is exercised

at node n, and, if exercised, the holder would have a payoff of hn. Auxiliary

variables, xn (free) and vn (nonnegative) are also introduced to denote that the

final wealth position can be unrestricted in sign. An additional auxiliary variable,


v, is introduced as the initial wealth of the portfolio.

(2.10)

max v

s.t.∑

n∈NT

pnxn − λ

√√√√√∑

n∈NT

pn

xn −

∑

k∈NT

pkxk

2

≥ 0

∑

m∈A(n)

em ≤ 1, n ∈ NT

zT0 θ0 = h0e0 − v

zTn (θn − θπ(n)) = hnen, n ∈ Nt, t = 1, . . . , T

zTn θn − xn − vn = 0, n ∈ NT

vn ≥ 0, n ∈ NT

en ∈ {0, 1}, n ∈ N.

The second-order cone constraints arise as risk constraints that provide a lower

bound for the Sharpe ratio of the final wealth position of the buyer. The term

∑

n∈NT

pnxn

is the expected value of the final wealth position,

∑

n∈NT

pn

xn −

∑

k∈NT

pkxk

2

is its variance, and λ is the lower bound on the Sharpe ratio.

The second set of constraints enforce that the option is exercised at no more

than 1 node in each sample path, and the remaining linear constraints ensure flow

balance through the scenario tree. The numerical results show that these prob-

lems, with over 20,000 continuous variables, 5,000 discrete variables, and 30,000


constraints to accommodate large enough scenario trees, are quite challenging for

existing MISOCP software.

2.2.3.2. Network Design and Operations

We now present a group of problems which we have loosely termed under the

heading of Network Design and Operations. They arise in telecommunications

networks that model the flow of commodities, cellular networks which must assign

base stations to mobile units, power systems, and highway networks with vehicular

traffic. Despite the similarities in the structures of the systems, the applications

all have different objectives and concerns, so there is a variety of different uses for

the binary variables and the second-order cone constraints in the following four

applications.

2.2.3.3. Delays in Telecommunication Networks

In [83], Hijazi et.al. investigate a telecommunications network problem that seeks

to minimize the network response time to a user request. The network is repre-

sented by vertices V and edges E, and vectors of capacities c and routing costs

w for the edges are given. We assume that the network can handle multiple com-

modities grouped by the set K, there is an amount vk of commodity k, and that

each commodity k has a set of candidate paths P (k), leading from its source to

its destination. The decision variables in the problem are continuous variables xe


representing the flow along edge e and φik representing the fraction of commodity

k routed along path Pik, as well as binary variables zik which indicate whether

the path Pik is open.

The initial model has the following form:

(2.11)

min∑

e∈E

wexe

s.t.∑

e∈Pik

1

ce − xe≤ αk, k ∈ K,Pik ∈ P (k) if zik = 1

∑

i:Pik∈P (k)

φik = 1, k ∈ K

∑

k∈K

(vk∑

Pik:e∈Pik

φik) = xe, e ∈ E

xe ≤ ce, e ∈ E

∑

Pik∈P (k)

zik ≤ N, k ∈ K

φik ≤ zik, k ∈ K,Pik ∈ P (k)

zik ∈ {0, 1}, k ∈ K,Pik ∈ P (k)

φik ≥ 0, k ∈ K,Pik ∈ P (k).

The objective function minimizes the total cost of all the flows along the edges

of the network. With ce denoting the capacity along edge e, the average queueing

plus transmission delay using an M/M/1 model is computed to be 1ce−xe

. Thus,

the first constraint ensures that the total end-to-end delay on any active path

through which a commodity k must travel is no greater than some parameter αk,

and it is this constraint that will require further examination. The second, third,

and fourth constraints ensure that all parts of a commodity are routed, that the


flow along each edge is the total flow over all the commodities that use the edge

as a part of one or more associated paths, and that the total flow along an edge

does not exceed the capacity of the edge. The fifth constraint states that the

commodity cannot be partitioned to more than N paths, and the sixth constraint

ensures that only the paths that will be opened for the commodity are allowed to

have flow of that commodity along them.

The authors re-examine the first constraint, and note that since the delay

along each open path is uncertain, they can also model it using a robust con-

straint. These constraints are also disjunctive since they are only used if the path

is open. To handle both the uncertainty and the disjunction, the authors propose

an extended formulation and a perspective function approach. The additional

details and notation required to introduce these MISOCPs is beyond the scope

of this paper, and the interested reader is referred to [83]. The numerical testing

shows that CPLEX has trouble solving large MISOCP instances, while related

MINLPs are solved in within reasonable time requirements by Bonmin.

2.2.3.4. Coordinated Multi-point Transmission in Cellular Networks

In [42], Cheng et.al. model and solve a coordinated multi-point transmission prob-

lem for cellular networks. For a network with L multiple-antenna base stations

and K single-antenna mobile stations, the problem is to find wkl as the beam-

forming vector used at base station l to transmit to mobile station k using the


following model

(2.12)

minK∑

k=1

L∑

l=1

(‖wkl‖2 + λklU(‖wkl‖))

s.t.L∑

l=1

U(‖wkl‖) ≤ ck, k = 1, . . . , K

SINRk ≥ γk, k = 1, . . . , KK∑

k=1

‖wkl‖2 ≤ Pl, l = 1, . . . , L,

where λkl denotes the penalty of serving mobile station l by base station k, the

function U(x) = 0 if x = 0 and 1 otherwise, ck is the maximum number of base

stations that can be assigned to mobile station k, SINRk is the receive signal-to-

interference-plus-noise ratio (SINR) at mobile station k, γk is the minimum SINR

level required to provide sufficient quality of service at k, and Pl is the maximum

available transmit power at base station l.

The SINR constraints can be reformulated as second-order cone constraints of

the form

‖(hHk W,σk)T‖ ≤

√1 + 1/γkRe{hHk wk}, Im{hHk wk} = 0, k = 1, . . . , K,

where hk represent the matrix of frequency-flat vectors to mobile station k, W is

the matrix whose columns are wk, k = 1, . . . , K, and σk is the standard deviation

of the white noise at mobile station k. In addition, binary variables akl and

auxiliary continuous variables tkl are introduced to convert (2.12) to the following


MISOCP:

(2.13)

minK∑

k=1

L∑

l=1

(tkl + λklakl)

s.t. ‖(2wTkl, akl − tkl)T‖ ≤ akl + tkl, k = 1, . . . , K, l = 1, . . . , LK∑

k=1

tkl ≤ Pl, l = 1, . . . , L

‖(hHk W,σk)T‖ ≤

√1 + 1/γkRe{hHk wk}, k = 1, . . . , K

Im{hHk wk} = 0, k = 1, . . . , KL∑

l=1

akl ≤ ck, k = 1, . . . , K

akl ∈ {0, 1}, k = 1, . . . , K, l = 1, . . . , L

tkl ≥ 0, k = 1, . . . , K, l = 1, . . . , L,

where the first set of second-order cone constraints are reformulations of the

quadratic constraints

‖wkl‖2 ≤ akltkl, k = 1, . . . , K, l = 1, . . . , L

as described in Subsection 2.1.1 for the portfolio optimization problem.

Due to the large size of the problem instances, the authors propose a heuristic,

which is able to obtain slightly worse solutions in significantly less CPU times

than CPLEX.


2.2.3.5. Power Distribution Systems

In [132], Taylor and Hover present several problems from power distribution sys-

tem reconfiguration, one of which is formulated as an MISOCP. Given a set of

lines W and a set of buses B, along with subsets W S ⊆ W with switches and

BF ⊆ B with substations, the goal is to minimize the loss by choosing the right

combination of open and closed switches along the system. The problem data

include the real and reactive powers from each substation i, pFi and qFi ; the real

and reactive loads at a bus i without a substation, pLi and qLi ; and resistance of

the line from bus i to j, rij . The model using the DistFlow equations of [9] has


the following form:

(2.14)

min∑

(i,j)∈W

rij(p2ij + q2

ij)

s.t.∑

k:(i,k)∈W

pik = pji − rijp2ji + q2

ji

v2j

− pLi , i ∈ B\BF

∑

k:(i,k)∈W

qik = qji − xijp2ji + q2

ji

v2j

− qLi , i ∈ B\BF

v2i = v2

j − 2(rijpji + xijqji) + (r2ij + x2

ij)p2

ji+q2ji

v2j

, (i, j) ∈ W

∑

j:(i,j)∈W

pij = pFi , i ∈ BF

∑

j:(i,j)∈W

qij = qFi , i ∈ BF

0 ≤ pij ≤ Mzij , (i, j) ∈ W

0 ≤ qij ≤ Mzij , (i, j) ∈ W

zij ≥ 0, (i, j) ∈ W

zif = 0, (i, f) ∈ W : f ∈ BF

zij + zji = 1, (i, j) ∈ W\W S

zij + zji = yij, (i, j) ∈ W S

∑

j:(i,j)∈W

zij = 1, i ∈ BF

yij ∈ {0, 1}, (i, j) ∈ W S

The decision variables are the continuous pij and qij denoting the real power flow

from bus i to j, continuous zij denoting the orientation of the line (i, j), and

the discrete yij denoting whether the switch on the line (i, j) will be open or


closed. In addition, the squared-variables v2i are the voltage magnitude. The first

three constraints represent the DistFlow equations, followed by two flow balance

constraints. The sixth and seventh constraints ensure that the power flow only

occurs along edges with open switches, and the remaining constraints seek to

define that power will flow only in one direction and in a manner consistent with

the network configuration.

When converting the problem into an MISOCP, the authors drop the last term

in the third constraint and replace the first three constraints with the following

system which includes auxiliary variables p, q, and v2:

∑

j:(i,j)∈W

(pij − pji) − pLi = pi, i ∈ B\BF

∑

j:(i,j)∈W

(qij − qji) − qLi = qi, i ∈ B\BF

v2i ≤ v2

j +M(1 − zji), (i, j) ∈ W

v2i ≥ v2

j −M(1 − zji), (i, j) ∈ W

rij(p2ji + q2

ji) ≤ v2i pi, (i, j) ∈ W

xij(p2ji + q2

ji) ≤ v2i qi, (i, j) ∈ W

v2i ≤ v2

j − 2(rijpji + xijqji) +M(1 − zij), (i, j) ∈ W

v2i ≥ v2

j − 2(rijpji + xijqji) −M(1 − zij), (i, j) ∈ W

While the authors do not mention doing so, we would also need to introduce

an auxiliary variable to move the quadratic objective function into a constraint

and then replace the constraint with an equivalent second-order cone constraint.


Numerical results are presented on 32 to 880 bus systems using CPLEX.

2.2.3.6. Battery Swapping Stations on a Freeway Network

In [101], Mak et.al. consider the problem of creating a network infrastructure

and providing coverage for battery swapping stations to service electric vehicles.

Given an existing freeway network, they consider candidate locations, J , and use

a binary variable, xj , for each candidate j to denote whether or not a swapping

station is located there. Additional binary variables, yjp and zjq denote whether

vehicles traveling along a path p ∈ P or a portion q ∈ Q of a path along the

network will visit swapping station j. The number of electric vehicles that travel

along each portion of a path is random, so demand at each swapping station is

uncertain. The model seeks to minimize the total cost, which consists of the

fixed costs associated with opening and operating the swapping stations and the


expected holding costs at each station.

(2.15)

min∑

j∈J

(fjxj + hGj(y))

s.t.∑

j∈J

ajqzjq ≥ 1, q ∈ Q

yjp ≥ bpqzjq, j ∈ J, p ∈ P, q ∈ Q

yjp ≤ xj, j ∈ J, p ∈ P

Hj(y) ≥ 1 − ǫ, j ∈ J

xj ∈ {0, 1}, j ∈ J

yjp ∈ {0, 1}, j ∈ J, p ∈ P

zjq ∈ {0, 1}, j ∈ J, q ∈ Q.

In the objective function, fj is the annualized fixed cost incurred if a station is

located at j ∈ J , and h is the annualized holding cost per battery. Gj(y) denotes

the expected largest total demand at swapping station j given the assignments

of stations to paths. If Q only contains those portions that are longer than a

maximum length dictated by battery life and denote by ajq a binary parameter

that indicates whether station j is on portion q, then the first constraint states

that there needs to be at least one swapping station along the portion q. In

addition, the second constraint states that, if portion q, with a station, is a part

of multiple paths as indicated by the binary parameter bpq with p ∈ P , then each

of those paths inherit the swapping station at q. The third constraint ensures

that vehicles are assigned only to stations that are open. In the fourth constraint,


Hj(y) is the worst-case probability of the demand at station j being less than the

number of simultaneous recharges permitted by the grid, and a worst-case service

level of at least 1 - ǫ is guaranteed, where ǫ > 0 is a small constant.

There are two parts of the problem, the nonlinear term in the objective function

and the chance constraint, that have to be dealt with before obtaining an MISOCP.

To handle the objective function term, auxiliary variables vj ≥ 0 are introduced

for each station j, modify the objective function to

∑

j∈J

(fjxj + hvj),

and let

vj ≥ Gj(y), j ∈ J.

It is shown in [101] that the worst-case scenario demand at j, Gj(y), has an upper

bound that consists of the sum of a Euclidean norm of a linear vector involving

y and another linear term also involving y. Due to the multitude of additional

notation in the calculation of this upper bound, we have not included the exact

formulation here and invite the interested reader to read the details in [101].

We simply note that such a construct leads to a second-order cone constraint.

The numerical studies conducted by the authors indicate that the upper bound

is accurate, and they also show that it is asymptotically tight if the underlying

uncertainties share the same descriptive statistics.

To handle the chance constraint indicating a robust service level requirement,


the authors introduce a Conditional Value-at-Risk constraint was used in our

earlier discussion on portfolio optimization problems. The resulting problem is

an MISOCP, and they solve it with data from the San Francisco freeway network

using CPLEX.

2.2.3.7. Euclidean k-center

In [31], Brandenberg and Roth introduce a new algorithm for the Euclidean k-

center problem, which deals with the clustering of a group of points among k balls

and arises in facility location and data classification applications. Without loss

of generality, assume that sets S1, . . . , Sk exist of points that are to be clustered

together and that there are still remaining points in S0 that have not yet been

assigned a cluster. There are a total of m points in Rn. The clusters, as stated,

will be enclosed in balls, and the continuous variables in the problem are the

coordinates of the centers, c ∈ Rn, for each ball. The binary variable, λij , denotes

the assignment of the points pj ∈ S0 to ball i. The model can be formulated as

follows:

(2.16)

min ρ

s.t. ‖pj − ci‖ ≤ ρ, pj ∈ Si, i = 1, . . . , k

‖λijpj − ci + (1 − λij)qij‖ ≤ ρ, pj ∈ S0, i = 1, . . . , kk∑

i=1

λij = 1, pj ∈ S0

λij ∈ {0, 1}, pj ∈ S0, i = 1, . . . , k.


The first set of second-order cone constraints is a reformulation of the requirement

to minimize the maximum Euclidean distance between a point and the center of

a cluster, and it is obtained by introducing an auxiliary variable ρ to denote the

maximum distance. The second set of second-order cone constraints are used to

denote that if a point is assigned to a ball, then the Euclidean distance between

the point and the center of the ball must be no more than the radius of the ball. If

the assignment is not made, then the constraint reduces to a given reference point

qij , already in the ball, being within the radius. This reference point is usually

chosen as the point in Si closest to pj .

We should note here that the authors consider norms other than the Euclidean

norm in the paper, and the MISOCP is a special case of their basic model. Nu-

merical results are obtained using a branch-and-bound method calling SeDuMi.

2.2.3.8. Operations Management

In [4], Atamturk et.al. explore a joint facility location and inventory management

model under stochastic retailer demand. The binary variables arise in the choice

of candidate locations at which to open distribution centers and the assignment of

retailers to the distribution centers. The second-order cone constraints appear in

the reformulation of the uncapacitated problem to move the nonlinear objective

function terms denoting the fixed costs of placing and shipping orders and the

expected safety stock cost into the constraints. The complete model has the


following form:

(2.17)

min∑

j∈J

(fjxj +

∑

i∈I

dijyij +Kjsj + qjtj

)

s.t.√∑

i∈I

µiy2ij ≤ sj, j ∈ J

√∑

i∈I

σ2i y

2ij ≤ tj , j ∈ J

∑

j∈J

yij = 1, i ∈ I

yij ≤ xj , i ∈ I, j ∈ J

xj ∈ {0, 1}, sj, tj ≥ 0, j ∈ J

yij ∈ {0, 1}, i ∈ I, j ∈ J,

where I is the set of existing retailers, J is the set of candidate locations for

opening distribution centers, and the variables x ∈ R|J | represent choices among

the candidates, with y ∈ R|I|×|J | assigning existing retailers to the new distribution

centers. Auxiliary variables are introduced to denote cost terms that are computed

nonlinearly

sj =√∑

i∈I

µiyij, tj =√∑

i∈I

σ2i yij,

and the fact that yij = y2ij is used to obtain the first two constraints in (2.17),

which are second-order cone constraints. In the otherwise linear problem, f is the

vector of fixed costs for opening a distribution center at each candidate location,

d is the matrix of unit shipping costs between retailers and distribution centers,

K and q aid in the calculation of costs for shipping, safety stocks, and any related

inventory costs for the assignments made to each distribution center, and µ and


σ denote the mean and standard deviation, respectively, of the daily demand at

each retailer. The third and fourth constraints in (2.17) ensure that each retailer

is assigned to only one distribution center and that assignment is only made to

those distribution centers which are open.

The model with capacities additionally has a second-order cone constraint

arising from moving an objective function term for the average inventory holding

cost into a constraint, and another one arising from the reformulation of a capacity

constraint. Other related models with similar features are provided in the paper.

Numerical results are conducted using algorithms studied in [124], [118], and

CPLEX.

2.2.3.9. Scheduling and Logistics

In [55], Du et.al. present an MISOCP as a relaxation of the MINLP that arises

in the problem of determining the berthing positions and order for a group of

vessels, V , waiting at a container terminal in order to minimize the total fuel cost


and waiting time of the vessels. The MINLP is formulated as follows:

(2.18)

min∑

i∈V

(c0i ai + c1

imµi

i a1−µi

i ) + λ∑

i∈V

(yi + hi − di)+

s.t. xi + li ≤ L, i ∈ V

xi + li ≤ xj + L(1 − σij), i, j ∈ V, i 6= j

yi + hi ≤ yj +M(1 − δij), i, j ∈ V, i 6= j

1 ≤ σij + σji + δij + δji ≤ 2, i, j ∈ V, i < j

ai ≤ ai ≤ ai, i ∈ V

ai ≤ yi, i ∈ V

ai, xi ≥ 0, i ∈ V

σij , δij i, j ∈ V, i 6= j

In (2.18), binary variables, σ and δ, are used to denote the relative positions of

pairs of vessels (whether one vessel is to the left of another and whether one vessel

is earlier than another). Additional continuous variables, x, a, and y, denote the

leftmost berthing positions, the terminal arrivals, and the start of the berthing

times for each vessel, respectively. In the problem data, L denotes the wharf

length at the terminal and l, h, and d denote the length, handling time, and

requested departure time of each vessel, respectively. As is customary, M denotes

an arbitrary large constant. The first four constraints of the problem are linear and

serve to set the rules on wharf length and the positions and handling time of each

vessel. Figure 1 from [55] depicts an example which clarify these relationships.

The fifth and sixth constraints allow the vehicle to adjust its sailing speed in order


to save fuel—the actual arrival time at the terminal is allowed to be in an interval

[ai, ai], while still remaining before the berthing time yi.

The MINLP model (2.18) incorporates two objective functions, fuel consump-

tion and total departure delay, both of which are minimized. We have introduced

a weight of λ in order to combine these two objective functions and simplify the

problem. Let us first discuss the fuel consumption objective function. This func-

tion is obtained using regression analysis for each vessel, and c0 and c1 denote the

regression coefficients, m denotes the distance of the vessels from the terminal,

and µi ∈ {3.5, 4, 4.5} for each i ∈ V . Introducing auxiliary variables q ∈ R|V |,

this function can be rewritten as

∑

i∈V

(c0iai + c1

imµi

i qi)

if the constraints

a1−µi

i ≤ qi, qi ≥ 0 i ∈ V

are introduced. These constraints can then be transformed into hyperbolic in-

equalities and then rewritten as second-order cone constraints. When µi = 3.5,

for example, additional variables ui1, ui2 ≥ 0 and the additional constraints

‖(2ui1, ai−1)‖ ≤ ai+1, ‖(2ui2, ui1−qi)‖ ≤ ui1+qi, ‖(2, ai−ui2)‖ ≤ ai+ui2

can be introduced. Similar transformations for µi = 4 and 4.5 are given in [55].

The second objective function is handled by introducing auxiliary variables


t ∈ R|V |, rewriting it as

∑

i∈V

ti,

and introducing the linear constraints

yi + hi − di ≤ ti, ti ≥ 0, i ∈ V.

With these transformations, the resulting problem is an MISOCP. Numerical

results are conducted for instances up to 28 vessels using CPLEX, which has

runtime and memory problems as the problem size grows.

2.3. Optimization Problems Arising in Portfolio Se-

lection

2.3.1. Markowitz Mean-Variance Framework

In the next two chapters, we extend the classical portfolio selection model de-

veloped by [103]. There are multiple reformulations of this model, in Chapter 3

we will focus on the case 1.2 where the investor’s objective function is to choose

the trading strategy to maximize expected return subject to constraints on the

maximum risk that the investor may be willing to take on, and in Chapter 4 we

will focus on the formulation that the investor’s objective function is to choose

the trading strategy to maximize expected return which is penalized variance-


covariance (market volatility) matrix:

(2.19)

maxw

rTw − λwTQw

s.t.∑ni=1 wi = 1

x ≥ 0

where Qij = Cov(Ri, Rj) is variance-covariance matrix of the vector of returns, R,

λ is risk aversion parameter. w is a vector of portfolio weights, and short-sale is

restricted with nonnegative weights in this model. We will consider the extended

version of (2.19) in both single and multi-period frameworks in Chapter 3.

In Chapter 4, we will consider two competing descriptions of portfolio selection,

the traditional mean variance efficient portfolio which is modeled as (1.2) versus

a generalization allowing for decision makers to consider skewness in their asset

allocation.

The classical mean-variance framework has been quite popular for portfolio

optimization problems since the 1960s. In [120], Phelps discusses an individual’s

optimal consumption policy using dynamic programming that maximizes wealth

under capital risk. In [112], Mossin provides optimal policies for both single-

and multi-period portfolio selection problems using dynamic programming tools.

Samuelson uses dynamic stochastic programming to find an optimal lifetime con-

sumption and investment policy in [123]. In [62], Fama also shows optimal con-

sumption policies for both single- and multi-period portfolio selection problems.

Hakansson has significant contributions to multi-period mean-variance portfolio


selection literature with [74], [75], [76], [77], [78]. Multi-period mean-variance

portfolio selection has also been studied in [60], [61], [64], [71], [94], [95], [96],

[129], [140], and [143].

2.3.2. Single- and Multi-Period Portfolio Optimization with

Cone Constraints and Discrete Decisions

Although there have been substantial developments in portfolio selection mod-

els during last six decades, there is still work to be done to incorporate more

complex components into these models and solve them efficiently. In this study,

we have chosen to incorporate transaction costs, conditional value-at-risk (CVar)

constraints, diversification requirements by sectors, and buy-in-thresholds into our

framework. The first two require the use of second-order cone constraints while

the latter two are implemented using binary variables, resulting in an MISOCP.

In addition, we further extend this model to the multiperiod case. These model

components/features have been adapted from [1], [29], [67], [73], and [99] as fol-

lows:

• According to [109], transaction costs include a number of factors, such as

price impacts of transactions, brokerage commissions, bid-ask spreads, and

taxes. About two decades ago, transaction costs started to be taken into

account by [48], [68], and [113] in portfolio optimization problems. In [141],


Yoshimoto models V-shaped transaction cost function in the mean-variance

portfolio optimization framework. However there are a number of different

ways to model transaction costs, including linear, piecewise linear and con-

vex or concave nonlinear cost functions. These transaction costs have also

been studied by [49], [56], [87], and [89] in a mean-variance framework.

We will use a quadratic cost function for the single-period model as proposed

in [67]:

1

2utΛut,

where ut = xt − xt−1, and Λ ∈ R(n+1)×(n+1) is the trading cost matrix and

is obtained as a positive multiple of the covariance matrix of the expected

returns. Because of this connection to a covariance matrix, Λ is symmetric

and positive definite. Note that both buy and sell transactions receive the

same transaction cost.

The multi-period portfolio optimization problem is obtained using a binary

scenario tree, and for this model we have to modify our transaction cost

function because of the fact that this quadratic convex formulation cause

non-convex algorithm, and for the multi-period model, we will use propor-

tional transaction cost in our model as Gurpinar et.al. do in [73] to preserve

convexity.

• We adopt our CVaR constraint from [99], where Lobo et.al. consider a


single-period portfolio selection problem with linear and fixed transaction

costs. They introduce a shortfall risk constraint in order to ensure that the

terminal wealth is greater than a predetermined threshold level. They obtain

second-order cone constraint from this formulation. We allow short-selling in

our model as they did. They consider specific portfolio optimization model

that is formulated as follows:

(2.20)

max αT (w + x+ − x−)

s.t. 1T (x+ − x−) +∑ni=1(a

+i x

+i + a−

i x−i ) ≤ 0

x+i ≥ 0, x−

i ≥ 0, i = 1, . . . , n

wi + x+i − x+

i ≥ si, i = 1, . . . , n

Φ−1(ηj)‖Σ1

2 (w + x+ − x−)‖ ≤ αT (wj + x+j − x−

j ) −W lowj , j = 1, . . . ,M

where αT ∈ Rn is the vector of expected returns on each asset, w ∈ Rn is

the vector of current holdings in each asset, and x ∈ Rn is the vector of

amounts transacted in each asset and Φ is the transaction cost function. By

using linear transaction costs, they obtain a convex optimization problem

which can be solved by using the general purpose software SOCP. When

they introduce fixed transaction costs into their framework, their model is no

longer convex. They relax their transaction cost constraint in order to obtain

a convex problem and solve the relaxed problem by using branch and bound

method. Their second approach is to provide an iterative heuristic. They


obtain a suboptimal solution with this method by solving a small number of

convex optimization problems. They show that there is a small gap between

the suboptimal heuristic solution and the guaranteed upper bound with

computational experiments. They suggest that these two methods can be

incorporated for further accuracy levels.

• In [29], Bonami and Lejeune study a single-period portfolio optimization

problem under stochastic and integer constraints as an extension of the clas-

sical mean-variance portfolio optimization framework. First they introduce

a probabilistic portfolio optimization model where expected asset returns

are stochastic and then they obtain their deterministic equivalents of these

models to test different probability distributions that can/can’t provide an

exact or approximate closed-form solution. They focus on different types of

constraints that traders should take into account when they are constructing

their portfolio, such as diversification by sectors, buy-in-threshold and round

lot constraints. The probabilistic Markowitz model with diversification-by-


sectors constraint is formulated as follows:

(2.21)

min wTΣw

s.t. µTw + F−1(1 − p)√wTΣw ≥ R

w0 +∑rj=1wj = 1

sminζk ≤ ∑j∈Sk

wk ≤ smin + (1 − smin)ζk, k = 1, . . . , L

∑Lk=1 ζk ≥ Lmin

ζ ∈ {0, 1}L

w ∈ Zr+1+

where F−1 is the inverse cumulative probability distribution of the portfolio

returns, Lmin is different economic sectors, and smin is the pre-determined

minimum value of investment level.

The probabilistic Markowitz model with buy-in-threshold constraint is for-

mulated as follows:

(2.22)

min wTΣw

s.t. µTw + F−1(1 − p)√wTΣw ≥ R

w0 +∑rj=1 wj = 1

wmin ≤ δj , j = 1, . . . , r

wminδj ≤ wj, j = 1, . . . , r

δ ∈ {0, 1}r

w ∈ Rr+1+


These constraints are studied by [85] in absence of uncertainty about a

decade ago. These sets of constraints provide binary and integer variables.

They use branch and bound algorithm with two new proposed branching

rules: Idiosyncratic risk and portfolio risk branching. Numerical results are

presented up to 200 assets with comparing standard MINLP solvers and [28].

They suggest that the portfolio risk branching rule performs best in terms

of robustness and speed. We adopt these formulations into our framework.

• In [73], Gurpinar et.al. introduce multi-period stochastic mean-variance

portfolio optimization problem. They include proportional transaction costs

in their model. The stochastic data is obtained by a scenario tree. They

obtain multistage stochastic quadratic programming model which is solved


by foliage, is a financial software package coded in C + +.

(2.23)

maxw,b,s

∑Tt=1 αt

∑e∈Nt

Pe[(wa(e) − wa(e))′(Λe + rer

′

e)(wa(e) − wa(e))]

s.t. p+ (1 − cb)b0 − (1 + cs)s0 = w0

1′b0 − 1′s0 = 1 − 1′p

re ◦ wa(e) + (1 − cb) ◦ be − (1 + cs) ◦ se = we, e ∈ NI

1′be − 1′se = 0, e ∈ NI

∑e∈NT

Pe[r′

e(wa(e) − wa(e))] ≥ W

wLe ≤ we ≤ wUe , e ∈ N

0 ≤ be ≤ bUe , e ∈ NI ∪ 0

0 ≤ se ≤ sUe , e ∈ NI ∪ 0

They test their model with WATSON dataset. Besides that they provide

computational backtesting experiments by using historical stock prices.

Although we are inspired from [67], [99], [29], and [73] when we build model,

we take forward all these studies in terms of comprehensiveness and complexity.

We consider real world portfolio constraints such as diversification by sectors, buy-

in thresholds constraints and total transaction costs. Although these constraints

provide meaningful financial interpretations to the portfolio, they result in a much

more complicated model. The overall model is a mixed-integer second-order cone

programming problem, a relatively new area of research. We consider this model

in both single and multi-period frameworks. We solve these model with a MAT-

LAB based Mixed Integer Linear and Nonlinear Optimization ([16]) solver that


implements a variety of methods for handling integer variables, cone constraints,

linear and nonlinear sub-problems. We have devised and implemented a solution

method for such problems and demonstrate its efficiency on large-scale portfolio

optimization models. We provide substantial improvement with warm-starting in

both branch-and-bound and outer approximation algorithms in terms of number

of iterations. I will discuss this study in Chapter 3.

2.3.3. Revealed Preferences for Portfolio Selection - Does Skew-

ness Matter?

Bilevel programming problem (BLPP) is a hierarchical optimization problem

where the constraint of an upper level optimization problem, is also an opti-

mization problem. In this framework, there are two independent decision makers,

leader and follower, who want to optimize their objectives. The leader moves first

and optimize her objective function with solving upper level optimization prob-

lem, and the follower observes the leader’s action and she moves sequentially to

optimize her objective function with solving lower level optimization problem with

given parameters of the upper level optimization problem. Therefore, this frame-

work is very similar with the Stackelberg leadership model which was proposed

in [138] by Von Stackelberg. In this model, leader firm, moves first and chooses

quantity where follower firm observes the leader’s action and chooses her quantity


which maximizes her profit. When both of the firms choose their quantities, then

market clearing price is set.

The general BLPP is formulated as follows:

minx∈X

F (x, y)

s.t. G(x, y) ≤ 0

H(x, y) = 0

miny∈Y

f(x, y)

s.t. g(x, y) ≤ 0

h(x, y) = 0

x, y ≥ 0

In this framework the leader moves first, to choose the optimal x vector to opti-

mize her objective function F (x, y). The follower observes this action and moves

sequentially to choose the optimal y vector to optimize her objective function

f(x, y).

Although the BLPP model first proposed by Bracken and McGill in [30], in [36]

Chandler and Norton first mentioned the term of "multilevel". Last five decades,

the BLPP models have been used to describe varies application problems in the

literature:

Agriculture: In [35], Chandler et al. study the potential role of multilevel pro-


gramming in agricultural economics. In [116], Onal et al. show that avail-

ability of the agricultural subsidy provide an increase in both aggregate

agricultural output and rural income by using bilevel programming model.

In [107], Miljkovic uses BLPP formulation to show the the effects of priva-

tization in YugoslaviaâŁ™s agricultural sector.

Economics: As we said before, bilevel programming problems subsume the Stack-

elberg duopoly model as discusses in [63]. In [125], Sherali et al. study the

existence and uniqueness of a Stackelberg-Nash-Cournot equilibrium in an

oligopoly model by using bilevel programming model. In addition bilevel

programming problems also study the principal-agent problem. In [134],

Ackere studies these problems in this context by analyzing a batch-size prob-

lem.

Engineering: In [44], Clark and Westerberg study BLPP for steady-state chem-

ical process design with thermodynamic equilibrium. BLP is also studied in

bioengineering. In [33], Burgard et al. use BLP framework to identify gene

knockout strategies for microbial strain optimization.

Government Policy: In [37], Cassidy et al. study the distribution of govern-

ment resources in this framework.

Management: In [10], Bard discusses coordination of decentralized organization


by using BLPP model. In [108], Miller et al. study facility location problem

under delivered pricing strategy. In [47], CÃ´tÃ© study airline revenue

management problem that solves the capacity allocation and pricing sub-

problems.

Transportation: BLPs are heavily used by [11], [43], [93], and [102] for network

design problems.

Please see [45], [51], [106], and [136] for more comprehensive literature review and

varies applications of BLPP models.

Although BLPP models are widely used in varies application as seen above,

there is only a couple of literature that related portfolio optimization. In [46],

Conn and Vicente study BLPP when both upper and lower level objective function

don’t have available derivative. They apply their derivative-free bilevel method

(Algorithm 5.2) to the robust optimization of the Omega function that is the ratio

of the weighted gains over the weighted losses. In the second study [97], Liou and

Yao introduce BLPP model for the mean-variance portfolio optimization problem.

They obtain an unique results for its MPEC (Mathematical Programming with

Equilibrium Constraints) problem formulation.


2.4. Optimization Problems Arising in Supply Chain

Management

Over the past several decades, there have been a variety of papers published on

the ELSP. Earliest works in this area include [58], [79], [122] and [105]. In these

studies, the lower bound (LB) for the ELSP solution was calculated using an

independent solution methodology, which ignored the sharing constraint and the

machine capacity issues. An improved LB approach was developed by [27], in

which the Karush-Kuhn-Tucker conditions were applied to the ELSP to account

for only the capacity constraint. Several researchers have utilized this LB for

comparative purposes ([111]). The bulk of the ELSP research has focused on

cyclic schedules which satisfy the Zero-Switch-Rule (ZSR), meaning an item is

produced only if its inventory depletes to a zero level. Nevertheless here are some

cases, such as [50] and, [105], where the ZSR was not considered.

As noted earlier, there are three approaches to solve the ELSP. The CC ap-

proach provides an upper bound to the optimal solution and yields very good

results under certain conditions ([65] and, [86]). The various heuristic methods

developed using the BP approach first selects a frequency for each item (i.e., the

number of times an item is produced in a production cycle). After the frequency

is determined, a basic time period to satisfy this frequency is then determined.

Earlier efforts along these lines include [27], [53]. In [59], Elmaghraby provided a


comprehensive review of this research. In [84], Hsu showed that using the basic

period approach to solve the ELSP is NP-hard and the NP-hardness increased

with an increase in the facility utilization ratio. Unlike the basic period approach,

the time-varying lot size approach does not require equal production runs. This

lot sizing approach was first examined by [105]. Subsequently, in [50], Delporte

and Thomas, and in [52], Dobson developed efficient heuristic techniques to show

that given enough time for production and set-up, any production sequence could

be converted into a feasible production schedule, although the timings and lot

sizes may not be equal. In [66], Gallego and Shaw provided support that the

time-varying lot size approach to the ELSP was generally NP-hard. More recent

explorations in this area has shown that Dobson’s heuristic in [52] can be inte-

grated with Zipkin’s optimum-seeking algorithm in [144], in order to generate near

optimal schedules in an efficient manner in [111].

In today’s competitive business environment, customers require dependable

on-time delivery at minimum cost from their suppliers and the ELSP can play a

key role in coordinating all the necessary activities of the various participants at

each stage of a supply network. However, as pointed out earlier, one characteris-

tic of much of the research on ELSP is that the finished products are consumed

at continuous rates. This implies that retail market demands for these products

are satisfied directly from the manufacturing facility. In today’s supply chains,

however, employing complex distribution networks, involving production plants,


vehicle terminals, airline hubs, warehouses, distribution centers, retail outlets,

etc., finished goods inventories from manufacturing plants are usually shipped

in bulk to succeeding stages along the distribution process. Moreover, existing

transport economies often tend to favor full truckload, rather that partial or less

than truckload shipments, in discrete, sizeable lots, for efficient movement of such

goods. Thus, it becomes necessary to re-examine the ELSP, with a focus on coordi-

nation and integration of the production schedule of a manufacturing process with

the transportation function, towards achieving greater supply chain efficiency.

In a review paper about the integrated analysis of production, distribution

and inventory planning, in [25], Bhatnagar et al. address the issue of coordina-

tion of activities in organizations. Two levels of coordination are discussed, i.e.

coordination of inter-organizational functions and coordination within the same

function at different echelons of an organization. In [40], Chandra and Fisher

align the production scheduling with the vehicle routing problem for examining

the value of coordination between these functions, employing a simulation study

of a two-echelon supply chain and a with one manufacturing plant and several

retailers.

In [26], Blumenfeld et al. and in [13], Benjamin consider multiple locations

within an echelon for the integrated analysis of production, inventory and trans-

portation decisions, under deterministic conditions. Blumenfeld et al. investigate

the trade-offs between transportation, inventory holding and production setup


costs in a supply chain. These authors analyze the cases of direct shipping be-

tween nodes, shipping through a terminal and a combination of both, and obtained

shipment sizes that consider the trade-offs between these costs. They are not con-

cerned with the capacity and the number of vehicles, but focus on obtaining the

value of the shipment size that trades off the respective costs. In [13], Benjamin

considers the simultaneous optimization of the production lot size, the transporta-

tion decision and the economic order quantity. He accounts for supply constraints

and explicitly considers inventory costs; his emphasis is on finding optimal pro-

duction batch sizes for supply points and order quantities for demand points. Our

analysis assumes an unconstrained transportation system and direct shipments

between nodes. Therefore, no routing issues were considered. Although this work

considers multiple products, no product to truck allocation decisions are made.

Generally speaking, coordination in production, inventory, and delivery has

been well addressed in the recent literature. There are a number of models on

integrated production, inventory and delivery decisions. In [142], Jonrinaldi and

Zhang proposed an integrated production and inventory model in an entire sup-

ply chain system, which consists of several raw materials and parts suppliers, a

manufacturer, multiple distributors and retailers. They propose a methodology

for determining integrated production and inventory cycles for multiple raw ma-

terials, parts, and products in a supply chain involving reverse logistics concepts.

A significant amount of recent research has focused on the area of linking produc-


tion scheduling and delivery activities, under various assumptions and objectives.

In [131], Lee and Yoon propose a coordinated production scheduling and delivery

batching model where different inventory holding costs are considered between

production and delivery stage. In [90], Georgios et al. formulate a mixed inte-

ger programming (MIP) model pertaining to the simultaneous food processing

and logistics planning problem for multiple products at various sites. In addition

to finding a feasible and optimal schedule, his proposed model help all the par-

ticipants in a supply chain collaborative process for obtaining the best balance

between production, inventory level, and distribution efficiency. Also, in a recent

study, in [98] Liu et al develop a multi-objective mixed-integer linear program-

ming (MILP) model with the minimization of total cost, total flow time, and

total lost sales as the objectives towards making optimal decisions with respect to

production, distribution, and capacity expansion.

72

Part I

Optimization Problems Arising in

Portfolio Optimization Models

CHAPTER 3. SINGLE- AND MULTI-PERIOD PORTFOLIO

OPTIMIZATION 73

Chapter 3

Single- and Multi-Period Portfolio

Optimization with Cone Constraints

and Discrete Decisions

3.1. Introduction

In this study, we extend the classical portfolio selection model developed by [103].

Although there are multiple reformulations of this model, we will focus only on the

case 1.2 where the investor’s objective function is to choose the trading strategy

to maximize expected return subject to constraints on the maximum risk that the

investor may be willing to take on.

In this chapter we focus on comprehensive MISOCP models that contain all


OPTIMIZATION 74

components/features proposed in [1], [29], [67], [73], and [99] that fit into our

framework. Our goal is to study how such models can be solved efficiently by

exploiting existing methods for MINLP and specialized approaches to solve the

underlying SOCPs.

The remainder of this chapter is organized as follows: The next section of this

chapter presents the single period portfolio optimization model. We discuss multi-

period portfolio optimization model and its formulation in depth in Section 3.3. In

Section 3.4, we propose two new algorithms for MISOCP, based on popular algo-

rithms for mixed-integer nonlinear programming (MINLP): a branch-and-bound

method and an outer approximation method. Both algorithms use a version of

the primal-dual penalty interior-point method proposed in [20] and [21] for solv-

ing the underlying SOCPs, which allows us to perform warmstarts and detect

infeasibilities in an efficient manner. In addition, we reformulate the second-order

cone constraint as discussed in [23] in order to convert the underlying SOCPs

into smooth convex nonlinear programming problems (NLP). Numerical testing

for these portfolio optimization problems have been conducted using the Matlab-

based solver MILANO [16] and are documented in Section 3.5. We will conclude

and discuss some future directions of this study in Chapter 6.


OPTIMIZATION 75

3.2. Single Period Portfolio Optimization Model

The single-period portfolio optimization model considered in this chapter can be

formulated as

(3.1)

maxx+,x−,ζ

rT (w + x+ − x−) − 1

2(x+ + x−)TΛ(x+ + x−)

s.t. Φ−1(ηk)‖Σ1

2 (w + x+ − x−)‖ ≤ rT (w + x+ − x−) −W lowk , k = 1, . . . ,M

sminζk ≤∑

j∈Sk

(wj + x+j − x−

j ) ≤ smin + (1 − smin)ζk, k = 1, . . . , L

L∑

k=1

ζk ≥ Lmin

n∑

j=0

(wj + x+j − x−

j ) = 1

wj + x+j − x−

j ≥ −sj , j = 1, . . . , n

x+, x− ≥ 0

ζ ∈ {0, 1}L,

where we consider cash (index 0) and n risky assets from L different sectors for

inclusion in our portfolio. The decision variables are x+ ∈ Rn+1 and x− ∈ Rn+1,

which denote the buy and sell transactions, respectively, and ζ ∈ {0, 1}L, the

elements of which denote whether there are sufficient investments in each sector.

We describe the remaining model components in detail below.


OPTIMIZATION 76

3.2.1. Objective Function

The investor’s objective is to choose the optimal trading strategies to maximize

the end-of-period expected total return. Denoting the expected rates of return

by r ∈ Rn+1 and the current portfolio holdings by w ∈ Rn+1, the expected total

portfolio value at the end of the period is given by

rT (w + x+ − x−).

However, both the buy and sell transactions are penalized by transaction costs.

According to recent dynamic portfolio choice literature ([67] and [32], for example),

transaction costs include a number of factors, such as price impacts of transactions,

brokerage commissions, bid-ask spreads, and taxes. As such, there are a number of

different ways to model transaction costs, including linear and convex or concave

nonlinear cost functions. In this work, we have decided to use the quadratic

convex transaction cost formulation of [67] as it provides best fit to our framework.

Therefore, the total transaction costs appear as a penalty term in the objective

function:

1

2(x+ + x−)TΛ(x+ + x−),

where Λ ∈ R(n+1)×(n+1) is the trading cost matrix and is obtained as a positive

multiple of the covariance matrix of the expected returns. Because of this con-

nection to a covariance matrix, Λ is symmetric and positive definite. Note that

both buy and sell transactions receive the same transaction cost, but it would be


OPTIMIZATION 77

straightforward to instead include two quadratic terms in the objective function

with different trading cost matrices for each type of transaction.

As we will see in the following discussion, the continuous relaxation of (3.1)

includes only linear and second-order cone constraints. However, the quadratic

term in the objective function prevents the overall problem from being formulated

as an MISOCP. While we could simply classify the problem as a MINLP, we

choose to instead reformulate it as an MISOCP so that the efficient algorithm we

will describe in the next section can be applied, allowing us to take advantage

of the special structure in the problem. We introduce a new variable ρ ∈ R and

rewrite the objective function of (3.1) as

n∑

j=0

rj(wj + x+j − x−

j ) − ρ,

with

1

2(x+ + x−)TΛ(x+ + x−) ≤ 2ρ.

Note that this constraint is equivalent to

(x+ + x−)TΛ(x+ + x−) ≤ (1 + ρ)2 − (1 − ρ)2,

and moving the last term to the left-hand side and taking the square root of both

sides gives the following second-order cone constraint:

∥∥∥∥∥∥∥∥∥

Λ1

2 (x+ + x−)

1 − ρ

∥∥∥∥∥∥∥∥∥≤ 1 + ρ.


OPTIMIZATION 78

Λ1

2 exists since Λ is positive-definite. Additionally, this conversion does not in-

crease the difficulty of solving the problem significantly—we add only one auxiliary

variable, so the Newton system does not become significantly larger. Also, wors-

ening the sparsity of the problem is not a concern here, since the original problem

(3.1) has a quite dense matrix in the Newton system due to the covariance matrix

and the related trading cost matrix both being dense.

3.2.2. Shortfall Risk Constraint

As stated above, we are considering both return and risk in this model. In the

objective function, we focus on maximizing the expected total return less transac-

tion costs, so we will seek to limit our risk using constraints. To that end, we will

use Conditional Value-at-Risk (CVaR) constraints, as was done by Lobo et.al. in

[99].

For each CVaR constraint k, k = 1, . . . ,M , we will require that our expected

wealth at the end of the period be above some threshold level W lowk with a prob-

ability of at least ηk. Thus, letting

W = rT (w + x+ − x−),

where r is the random vector of returns, we require that

P(W ≥ W lowk ) ≥ ηk, k = 1, . . . ,M.


OPTIMIZATION 79

We assume that the elements of r have jointly Gaussian distribution so that

W is normally distributed with a mean of

rT (w + x+ − x−)

and a standard deviation of

‖Σ1

2 (w + x+ − x−)‖,

where Σ is the covariance matrix of the returns.

Therefore, the CVaR constraints can be formulated as

P(W − rT (w + x+ − x−)

‖Σ1

2 (w + x+ − x−)‖≥ W low

k − rT (w + x+ − x−)

‖Σ1

2 (w + x+ − x−)‖

)≥ ηk,

for each k = 1, . . . ,M . This implies that

1 − Φ

(W lowk − rT (w + x+ − x−)

‖Σ1

2 (w + x+ − x−)‖

)≥ ηk, k = 1, . . . ,M,

where Φ is the cumulative distribution function for a standard normal random

variable, or

Φ(z) =1√2π

∫ z

−∞e−t2/2dt.

Rearranging the terms and taking the inverse gives us

W lowk − rT (w + x+ − x−)

‖Σ1

2 (w + x+ − x−)‖≤ Φ−1(1 − ηk), k = 1, . . . ,M.

Using the symmetry of the standard normal distribution function, we can rewrite

the constraint again as

−W lowk − rT (w + x+ − x−)

‖Σ1

2 (w + x+ − x−)‖≥ Φ−1(ηk), k = 1, . . . ,M.


OPTIMIZATION 80

Finally, rearranging the terms gives us the second-order cone constraint in (3.1)

Φ−1(ηk)‖Σ1

2 (w + x+ − x−)‖ ≤ rT (w + x+ − x−) −W lowk , k = 1, . . . ,M.

3.2.3. Diversification By Sectors

Diversification is another important instrument used to reduce the level of risk in

the portfolio. In this part, we impose a diversification requirement to the investor

to allocate sufficiently large amounts in at least Lmin of the L different economic

sectors. This type of constraint was considered by [29].

To express this diversification requirement, we start by defining binary vari-

ables ζk ∈ {0, 1}, k = 1, . . . , L for each economic sector k. If ζk = 1, our total

portfolio allocation in assets from sector k will be at least smin (and, of course, no

more than 1). Otherwise, it will mean that our total portfolio allocation in those

assets fell short of the threshold level smin. We can express these requirements

with a constraint in the following form:

sminζk ≤∑

j∈Sk

(wj + x+j − x−

j ) ≤ smin + (1 − smin)ζk, k = 1, . . . , L,

where Sk is the set of assets that belong to economic sector k, k = 1, . . . , L.

In order to express the diversification requirement, we also need to introduce

a cardinality constraint:

L∑

k=1

ζk ≥ Lmin.(3.2)


OPTIMIZATION 81

3.2.4. Portfolio Constraints

The remaining constraints in our problem are grouped into the general category

of portfolio constraints. The first of these,

n∑

j=0

(wj + x+j − x−

j ) = 1,

requires that we allocate 100% of our portfolio at the end of the investment period.

Since we start withn∑

j=0

wj = 1,

this constraint can also be written as

n∑

j=0

x+j =

n∑

j=0

x−j ,

which provides a balance between the buy and sell transactions.

Additionally, we have another constraint that allows for shortsales of the

nonliquid assets by stating that we can take a limited short position for each

one:

wj + x+j − x−

j ≥ −sj , j = 1, . . . , n,

where s represents the short position limit for each nonliquid asset.

Finally, we require that x+ and x−, the variables associated with the buy and

sell transactions must be nonnegative.

With the modifications to the model due to the transaction costs, the MISOCP


OPTIMIZATION 82

we will be solving in our numerical testing will have the form

(3.3)

maxx+,x−,ζ,ρ

rT (w + x+ − x−) − ρ

s.t.

∥∥∥∥∥∥∥∥∥

Λ1

2 (x+ + x−)

1 − ρ

∥∥∥∥∥∥∥∥∥≤ 1 + ρ

Φ−1(ηk)‖Σ1

2 (w + x+ − x−)‖ ≤ rT (w + x+ − x−) −W lowk , k = 1, . . . ,M

sminζk ≤∑

j∈Sk

(wj + x+j − x−

j ) ≤ smin + (1 − smin)ζk, k = 1, . . . , L

L∑

k=1

ζk ≥ Lmin

n∑

j=0

(wj + x+j − x−

j ) = 1

wj + x+j − x−

j ≥ −sj , j = 1, . . . , n

x+, x− ≥ 0

ζ ∈ {0, 1}L.

As mentioned above, there are two additional types of constraints appearing in

literature that we would like to include in future testing. The first of these, buy-in-

threshold constraints, require additional binary variables and linear constraints,

and therefore keep the problem as an MISOCP. We did not include them in this

study since we already have an MISOCP and the particular data set we chose

led to either infeasible or trivially solved problems when the buy-in-thresholds

were added. The second type of constraint, round-lot constraints, could introduce

nonlinear functions into our constraints, and we wanted to focus on MISOCPs


OPTIMIZATION 83

in this chapter and leave MINLPs with second-order cone constraints for future

work. For completeness, however, we include a brief description of both of these

types of constraints.

3.2.5. Buy-in-Threshold Constraints

Since we have included transaction costs in our model (3.3), we will be mindful

of the number of transactions, as well. Therefore, we can impose a requirement

that the investors do not hold very small active positions (see [29]). Introducing

new binary variables δ ∈ {0, 1}n, we can write this requirement using constraints

of the following form:

wminδj ≤ wj + x+j − x−

j ≤ δj , j = 1, . . . , n,

where wmin is a predetermined proportion of the available capital.

3.2.6. Round-Lot Constraints

For certain types of investments, such as real estate, we might be required to hold

an integer number of assets. Therefore, we could consider adding a constraint of

the form

wj + x+j − x−

j =pjγjMj∑n

k=0 pk(wk + x+k − x−

k )j = 1, . . . , n,

where pj is the face value of one unit of asset j, γj ∈ Z is the (nonnegative integer)

decision variable denoting the number of assets held of that type, and Mj is the


OPTIMIZATION 84

batch size for the asset.

3.3. Multi-Period Portfolio Optimization Model

In this section we consider the multi-period portfolio optimization problem. There

are multiple ways to formulate and solve multi-period portfolio optimization prob-

lems in literature, such as dynamic programming [38] and robust optimization [24].

In this study, we obtain the multi-period model by constructing a scenario tree.

These model features were adapted from [73].

3.3.1. Scenario Tree

The use of scenario trees is not a new concept for multi-period financial/portfolio

optimization problems. In multi-period portfolio allocation literature, the gener-

ation of scenario trees is discussed in [72] and [119], and scenario trees studied

in [34] and [73]. We construct a binary scenario tree in Figure 1 to illustrate

some of the important concepts and notation. There are discrete decision periods

t = 1, . . . , T at which to reallocate the volumes of n risky assets and a riskless

asset in the portfolio. N represents the set of all nodes in the scenario tree, e ∈ N

represents the index of the event (s, t), the ordered pair of scenario s and time

period t. The parent node of e in the scenario tree is denoted by a(e). The

branching probability is denoted by Pe = Πi=1,...,tp(s,i).


OPTIMIZATION 85

Figure 3.1: Scenario Tree

3.3.2. Objective Function

The investor’s objective function is to choose the optimal trading strategies (xe =

x+e − x−

e ), to maximize the-end-of period expected return. The expected rate of

return is denoted by re ∈ Rn+1 and the current portfolio holdings are denoted by

we ∈ Rn+1 for event e. The end-of-period expected return is formulated as:


OPTIMIZATION 86

WT = E[rT (ξT )wT−1]

= E[rT (ξT |ξT−1)wT−1]

= E

∑

e∈NT

Per⊤Twa(e)

=∑

e∈NT

Perewa(e)

where ξt is the stochastic data at time t, ξt represents historic data up to time t

and re is the stochastic realization of rT (ξT |ξT−1).

3.3.3. Transaction Costs Constraints

In a multi-period framework, we assume that transaction costs are paid on a

period-by-period basis. As such, the payment of transaction costs needs to be

incorporated into the flow balance constraints, which are modeled an equalities.

If we were to keep the quadratic transaction costs of the single-period framework,

the nonlinearities in the flow balance constraints would lead to the MINLP hav-

ing nonconvex nonlinear relaxations. Therefore, we have decided to use linear

transaction costs for the multi-period model and leave the quadratic case for fu-

ture work on MINLPs. Such linear transaction costs accurately model brokerage

commissions on transacted assets.

We impose transaction cost for both buying (cb) and selling (cs) strategies.


OPTIMIZATION 87

Therefore, we obtain following balance constraint:

we = re ◦ wa(e) + x+e ◦ (1 − cb) − x−

e ◦ (1 + cs), ∀e ∈ NI

We also require that

1⊤x+e = 1⊤x−

e ∀e ∈ NI

where NI represents the set of all interior nodes of the scenario tree.

3.3.4. Shortfall Risk Constraints

To model shortfall risk, we follow the same procedure as in the single period model

in Section 3.2.2. This constraint provides a requirement that the end-of-period

wealth W stay above of some undesired level W low with a probability greater than

η. Therefore, we can formulate the end-of-period shortfall risk constraint using

the following steps:

Letting

W =∑

e∈NT

Perewa(e),

where re is the stochastic realization of rT (ξT |ξT−1), we require that

W ≥ W lowk ≥ ηk, k = 1, . . . ,M.

We assume that the elements of r have jointly Gaussian distribution so that

W is normally distributed with a mean of

∑

e∈NT

Perewa(e)


OPTIMIZATION 88

and a standard deviation of

‖Σ1

2we‖,

where Σ is the covariance matrix of the returns. As shown in [29], we can expand

this assumption to a more general class of probability distributions, including

symmetric probability distributions and positively skewed probability distribu-

tions. In fact, [29] also shows that our proposed model can approximate an even

greater class of distributions, which encompasses any distribution that can be

characterized by its first two moments.

Therefore, the CVaR constraints can be formulated as

P(W − rewa(e)

‖Σ1

2we‖≥ W low

k − rewa(e)

‖Σ1

2we‖

)≥ ηk,

for each k = 1, . . . ,M . This implies that

1 − Φ

(W lowk − rewa(e)

‖Σ1

2we‖

)≥ ηk, k = 1, . . . ,M.

Rearranging the terms and taking the inverse gives us

W lowk − rewa(e)

‖Σ1

2we‖≤ Φ−1(1 − ηk), k = 1, . . . ,M.

Using the symmetry of the standard normal distribution function, we can rewrite

the constraint again as

−W lowk − rewa(e)

‖Σ1

2we‖≥ Φ−1(ηk), k = 1, . . . ,M.


OPTIMIZATION 89

Finally, rearranging the terms gives us the second-order cone constraint in (3.1)

Φ−1(ηk)‖Σ1

2we‖ ≤ rewa(e) −W lowk , k = 1, . . . ,M.

In addition, we include the following to impose risk constraints for the interior

branches of the tree:

we ≥ 0.85wa(e) e = 1, . . . , T − 1

Using different constraints on wealth allows us to be less conservative in the in-

terior branches, where we might take a slight loss in one period to realize bigger

gains later.

3.3.5. Diversification By Sectors

We already discussed this constraint in Section 3.2.3 in the single period model.

In this part, we just manipulate the constraints to satisfy the multi-period setting.

Therefore we obtain the following set of constraints.

sminζke ≤∑

i∈Sk

we ≤ smin + (1 − smin)ζke, k = 1, ..., L

L∑

k=1

ζke ≥ Lmin, ζ ∈ {0, 1}L×N


OPTIMIZATION 90

3.3.6. Portfolio Constraints

We manipulate the portfolio constraints that are discussed in Section 3.2.4 to

apply them in multi-period framework:

x+e ≥ 0, x−

e ≥ 0, we ≥ −s, ∀e ∈ N .

3.3.7. Buy-in-Threshold Constraints

In this section, we adopt the constraints introduced in Section 3.2.5 to the multi-

period setting:

wminδe ≤ we ≤ δe, ∀e ∈ N and δ ∈ {0, 1}n×N .

We obtain the following MISOCP prbolem for the multi-period portfolio op-

timization problem:


OPTIMIZATION 91

maxwe,x

+e ,x

−

e ,ζe,δe

∑

e∈NT

Perewa(e)

s.t. we = re ◦ wa(e) + x+e ◦ (1 − cb) − x−

e ◦ (1 + cs), ∀e ∈ NI

1⊤x+e = 1⊤x−

e ∀e ∈ NI

Φ−1(ηj)‖Σ1

2we‖ ≤ rewa(e) −W lowj , j = 1, . . . , m, ∀e ∈ NT

we ≥ 0.85wa(e) ∀e ∈ NI

sminζke ≤∑

i∈Sk

we ≤ smin + (1 − smin)ζke, k = 1, ..., L, ∀e ∈ N

L∑

k=1

ζke ≥ Lmin, ∀e ∈ N

ζ ∈ {0, 1}L×N

we ≤ δe, ∀e ∈ N

wminδe ≤ we, ∀e ∈ N

δ ∈ {0, 1}n×N

x+e ≥ 0, ∀e ∈ N

x−e ≥ 0, ∀e ∈ N

we ≥ −s, ∀e ∈ N


OPTIMIZATION 92

3.4. Solving the MISOCP

In this section, we will describe two MINLP approaches that we have adapted

for MISOCP. As stated, there are three important issues to consider: nondif-

ferentiability of the underlying SOCP, warmstarting when solving a sequence of

SOCPs, and infeasibility detection. We will address the first using a smooth con-

vex reformulation of the SOCP and the latter two using a primal-dual penalty

interior-point method.

3.4.1. The Ratio Reformulation

In [23], Benson and Vanderbei investigated the nondifferentiability of an SOCP

and proposed several reformulations of the second-order cone constraint to over-

come this issue. Note that the nondifferentiability is only an issue if it occurs at

the optimal solution. Since an initial solution can be randomized, especially when

using an infeasible interior-point method to solve the SOCP, the probability of

encountering a point of nondifferentiability is 0.

For a constraint of the form

(3.4) ‖u‖ ≤ t

where u is a vector and t is a scalar, Benson and Vanderbei proposed the following:

• Exponential reformulation: Replacing (3.4) with e(uT u−t2)/2 ≤ 1 and t ≥ 0


OPTIMIZATION 93

gives a smooth and convex reformulation of the problem, but numerical

issues frequently arise due to the exponential.

• Smoothing by perturbation: Introducing a scalar variable v into the norm

gives a constraint of the form√v2 + uTu ≤ t, but in order for the formulation

to be smooth, we need v > 0. This is ensured by setting v ≥ ǫ for a small

constant ǫ, usually taken around 10−6 − 10−4.

• Ratio reformulation: Replacing (3.4) with uT ut

≤ t and t ≥ 0 yields a

convex reformulation of the problem, but the constraint function may still

not be smooth. Nevertheless, in many applications, such as the portfolio

optimization problems to be studied in the next section, the right-hand side

of the second-order cone constraint in (1.1) is either a scalar or bounded

away from zero at the optimal solution.

While the exponential reformulation and smoothing by perturbation resolve

the nondifferentiability issue for the general SOCP, the ratio reformulation will be

our pick for this study since we will focus only on portfolio optimization problems.

The numerical issues due to the exponential function were causing failures during

our numerical studies, and the smallest lower bounds that would avoid numerical

nondifferentiability were still too big for the scale of the numbers in the second-

order cone constraints in our problems. Applying the ratio reformulation, we will


OPTIMIZATION 94

be solving the following MISOCP instead of (1.1)

(3.5)

minx∈X

cTx

s.t.(Aix+ bi)

T (Aix+ bi)

aT0ix+ b0i≤ aT0ix+ b0i, i = 1, . . . , m

aT0ix+ b0i ≥ 0, i = 1, . . . , m.

We picked the ratio reformulation in order to guarantee that the underlying

SOCPs would be smooth. We will now examine this choice for the second-order

cone constraints included in (3.3).

For the transaction cost constraints, the right-hand side term is 1 + ρ. Since

the total transaction cost paid will be 2ρ, we have that ρ ≥ 0. Therefore, 1+ρ ≥ 1,

and the right-hand side is bounded away from 0.

For the shortfall constraints, note that we start with∑nj=0 wj = 1 and that,

since we are focusing on shortfalls, W lowk < 1. Also note that if we assume that

our initial asset allocation satisfies the diversification by sector constraints, we

can define a feasible solution that does not require us to buy or sell any assets.

Our objective is to maximize our end-of-period expected total return, which means

that we expect our optimal allocation to do at least as well as this feasible solution.

Thus, we can guarantee that

n∑

j=0

rj(wj + x+j − x−

j ) ≥ 1 > W lowk , k = 1, . . . ,M,

which means that the right-hand side is bounded away from 0.

With these reformulations, the first two constraints in (3.3) can be rewritten


OPTIMIZATION 95

as

(x+ + x−)T Λ(x+ + x−) + (1 − ρ)2

1 + ρ≤ 1 + ρ

(Φ−1(ηk))2(w + x+ − x−)T Σ(w + x+ − x−)

rT (w + x+ − x−) −W lowk

‖ ≤ rT (w + x+ − x−) −W lowk

, k = 1, . . . ,M.

3.4.2. The Primal and Dual Penalty Problems

In order to solve the SOCPs that will arise during the course of the branch-and-

bound and the outer approximation methods, we will use the primal-dual penalty

interior-point method that was introduced in [20] for linear programming and in

[21] for nonlinear programming. This approach includes relaxation/penalty terms

in both the primal and the dual problems, which imbues the algorithm with the

ability to perform warmstarts and detect infeasibilities. The new terms do not

change the structure of the problem, that is, we will still solve an SOCP and can

continue to use a highly efficient interior-point method to do so. In addition, the

relaxation scheme creates strict interiors for the feasible regions of both the primal

and the dual problems, thereby providing a regularization and allowing for the

solution of SOCPs that may not otherwise satisfy standard assumptions for the

interior-point method to work.

Even though our approach is to solve the SOCP as a nonlinear programming

problem, the particular relaxation/penalty scheme differs slightly from the one

presented in [21]. If we were to follow the outline of the approach presented in


OPTIMIZATION 96

that paper, the relaxed SOCP constraint would have the form

(Aix+ bi)T (Aix+ bi)

aT0ix+ b0i≤ aT0ix+ b0i + ξi

aT0ix+ b0i + ρi ≥ 0

ξi, ρi ≥ 0,

where ξ, ρ ∈ Rm are the relaxation variables that would get penalized in the ob-

jective function. While this would provide a sufficient relaxation for our purposes,

we have decided to use the following relaxation instead:

(3.6)

(Aix+ bi)T (Aix+ bi)

aT0ix+ b0i + ξi≤ aT0ix+ b0i + ξi

aT0ix+ b0i + ξi ≥ 0

ξi ≥ 0,

This form of the relaxation can be obtained in two different ways:

• If we apply the relaxation scheme from [21] to the second-order cone con-

straint in (1.1), we obtain

‖Aix+ bi‖ ≤ aT0ix+ b0i + ξi, ξi ≥ 0.

Note that we have a second-order cone and a linear constraint after the

relaxation. If we apply the ratio reformulation now, we obtain (3.6).

• The ratio reformulation constraint in (3.5) can also be written as a semidef-

inite constraint of the form

(aT0ix+ b0i)I Aix+ bi

(Aix+ bi)T aT0ix+ b0i

� 0,


OPTIMIZATION 97

where I is the mi ×mi identity matrix. As outlined in [23], the semidefinite

constraint is equivalent to the entries of the diagonal matrix D in the LDLT

factorization of the above matrix being nonnegative. Without permutation,

we have that

Djj =

aT0ix+ b0i, j = 1 . . .mi

aT0ix+ b0i − (Aix+ bi)T (Aix+ bi)

aT0ix+ b0i

, j = mi + 1,

so Djj ≥ 0 for j = 1, . . .mi + 1 matches the constraints in (3.5). The

semidefinite constraint can be relaxed by adding a positive definite diagonal

matrix to the left-hand side:(aT0ix+ b0i)I Aix+ bi

(Aix+ bi)T aT0ix+ b0i

+ ξiI � 0, ξi ≥ 0,

where I is the (mi+1)×(mi+1) identity matrix. The first two inequalities in

(3.6) correspond to nonnegativity requirements on the entries of the diagonal

matrix in the LDLT factorization of this matrix. The third inequality in

(3.6) exactly matches the nonnegativity of ξ to ensure that this is indeed a

relaxation.

One advantage of this relaxation formulation over the one presented in [20] and

[21] is that we only use m relaxation variables instead of 2m. Doing so means

that we will have not only fewer variables but also fewer penalty parameters to

control in the resulting primal-dual penalty problem.


OPTIMIZATION 98

Thus, the primal penalty problem can be formulated as

(3.7)

minx,ξ

cTx+ dT ξ

s.t.(Aix+ bi)

T (Aix+ bi)

aT0ix+ b0i + ξi≤ aT0ix+ b0i + ξi, i = 1, . . . , m

aT0ix+ b0i + ξi ≥ 0, i = 1, . . . , m

aT0ix+ b0i ≤ ui, i = 1, . . . , m

ξi ≥ 0, i = 1, . . . , m,

where d and u are the strictly positive primal and dual penalty parameters, respec-

tively. As discussed in [20] and [21], relaxing a constraint in the primal problem

leads to the primal penalty parameter of the relaxation acting as an upper bound

on the dual variables. In order to establish a similar relaxation on the dual side,

we introduce an upper bound on the primal side, and, again, this upper bound

ends up serving as the dual penalty parameter of the dual relaxation. In fact, the

dual problem has the following form:

(3.8)

maxy0,y,ψ

−m∑

i=1

(bTi yi + b0iy0i + uiψi)

s.t.m∑

i=1

(ATi yi + a0iy0i) = c

y0 + ψ ≤ d

y0 + ψ ≥ 0

yTi yiy0i + ψi

≤ y0i + ψi, i = 1, . . . , m

ψ ≥ 0,

where yi ∈ Rmi , i = 1, . . . , m and y0 ∈ Rm are the dual variables and ψ ∈ Rm


OPTIMIZATION 99

are the dual relaxation variables.

Note that for sufficiently large d and u, both (3.7) and (3.8) have strictly

feasible interiors. For the primal problem, we can pick any x, set u to satisfy

aT0ix+ b0i < ui for i = 1, . . . , m, and we can let

ξi > max{0,−(aT0ix+ b0i), ‖Aix+ bi‖ − (aT0ix+ b0i)}.

Similarly, for the dual problem, pick any y and set y0 in order to satisfy the first

constraint of (3.8). (Since we no longer require y0 ≥ 0, it is possible to do so.)

Then, we can pick any

ψi > max{0,−y0i, ‖yi‖ − y0i}

and set di > y0i + ψi for i = 1, . . . , m.

Having strictly feasible interiors for both the primal and the dual problems

means that both (3.7) and (3.8) have optimal solutions, and there is no duality

gap. Thus, the pair (3.7) and (3.8) satisfy the regularity assumptions of standard

interior-point algorithms for both SOCP and general NLP ([2], [19]).

Nevertheless, even though (3.7) and (3.8) exhibit regularity, the original SOCP

may not. In fact, as it quite often happens within a branch-and-bound framework,

the original SOCP may not even be feasible. It is shown in [19] that a solution

with a duality gap, if it exists, can be recovered as the penalty parameters (either

the primal or the dual, while keeping the other fixed) tend to infinity. Similarly, it

is well-known that the original objective function can be dropped and a feasibility


OPTIMIZATION 100

problem can be solved as needed. One advantage of having a relaxation/penalty

scheme for both the primal and the dual problems is that a feasibility problem

can be designed for either one, in order to detect primal or dual infeasibility for

the original SOCP.

3.4.3. A Primal-Dual Penalty Interior-Point Method

Since we will solve the pair (3.7)-(3.8) as NLPs, we will now describe the ap-

plication of a standard interior-point method to these problems. This method,

along with approaches to manage the penalty parameters, has been discussed ex-

tensively in [21] for a general NLP, so we will only provide a brief outline here,

adapted to the case of a reformulated SOCP. Since the relaxed constraint (3.6)

looks slightly different than the relaxed constraint in [21], we will need to intro-

duce the appropriate first-order conditions, but the general outline of the overall

solution method will be the same.

We start by introducing some auxiliary variables that will help simplify our


OPTIMIZATION 101

formulation:

(3.9)

minx,ξ,f,g

cTx+ dT ξ

s.t. fi = Aix+ bi, i = 1, . . . , m

gi = aT0ix+ b0i + ξi, i = 1, . . . , m

gi − fTi figi

≥ 0, i = 1, . . . , m

gi ≥ 0, i = 1, . . . , m

ui − gi + ξi ≥ 0, i = 1, . . . , m

ξi ≥ 0, i = 1, . . . , m,

where fi ∈ Rmi and gi ∈ R, i = 1, . . . , m are the auxiliary variables. Since

the first two constraints that serve to introduce these variables are affine equality

constraints, (3.9) remains a convex nonlinear programming problem.

Formulating the log-barrier problem for (3.9):

(3.10)

minx,ξ,f,g

cTx+ dT ξ − µm∑

i=1

(log

(gi − fTi fi

gi

)+ log gi + log(ui − gi + ξi) + log ξi

)

s.t.fi = Aix+ bi, i = 1, . . . , m

gi = aT0ix+ b0i + ξi, i = 1, . . . , m,

where µ > 0 is the barrier parameter.


OPTIMIZATION 102

The first-order conditions for this problem are:

(3.11)

Aix− fi + bi = 0, i = 1, . . . , m

aT0ix+ ξi − gi + b0i = 0, i = 1, . . . , m

c−m∑

i=1

ATi yi −m∑

i=1

a0iy0i = 0

ψi(ui − gi + ξi) = µ, i = 1, . . . , m

ξi(di − y0i − ψi) = µ, i = 1, . . . , m

(y0i + ψi)

(gi − fTi fi

gi

)= µ, i = 1, . . . , m

y0i + ψigi

fi + yi = 0, i = 1, . . . , m.

Note that the last condition implies the second-order cone constraint in (3.8) since

we would have that

yTi yi = (y0i + ψi)2 f

Ti fig2i

and fTifi

g2i

≤ 1 in each iteration.

The first-order conditions are solved using Newton’s Method while performing

a linesearch to guarantee progress toward optimality and modifying the value of

µ at each iteration (see [21] or [22] for details). Of course, we need to also control

the penalty parameters to guarantee that we have found a solution for the original

SOCP or provide a certificate of infeasibility. In [21], Benson and Shanno discuss

two approaches, static and dynamic updating, to resolve this issue.

• For static updating, the values of d and u are kept constant, and the problem

is solved to optimality. Then, if ξ > 0 (or ψ > 0) at the optimal solution, the


OPTIMIZATION 103

primal (or the dual) penalty parameters are increased and the new problem

is solved. After a fixed number of updates are performed, the problem is

declared a candidate for infeasibility. If another update is necessary, c is set

to 0 before solving the system again to detect primal infeasibility (or b is

set to 0 to detect dual infeasibility). If a feasible solution (for the original

SOCP) is obtained at the end of this process, we return to solving (3.11)

with higher values of the penalty parameters. Otherwise, we declare the

problem to be infeasible.

• For dynamic updating, the progress of gi + ξi and y0i + ψi for i = 1, . . . , m

toward their upper bounds of ui and di, respectively, are monitored at each

iteration. If any of them are too close to their upper bounds, those bounds

are increased. If any single bound is increased more than a fixed number of

times, we modify the corresponding problem as described in static updating

to enter the infeasibility detection phase. Similarly, if a feasible solution is

found, we return to solving the original problem. Otherwise, we declare the

problem to be infeasible.

While the static update is rather straightforward, it may require the complete

solution of multiple problems. Therefore, as was the case in [21], the dynamic

updating approach is preferred here as well.

In addition to its warmstarting capabilities, the primal-dual penalty approach


OPTIMIZATION 104

also allows us to (approximately) solve SOCPs that have duality gaps at the

optimal pair of primal-dual solutions. This asymptotic behavior of the relaxed

problem is analyzed in [19].

3.4.4. Warmstarting

Most successful implementations for mixed-integer linear programming either use

a simplex-type method to solve the underlying linear programming problems, or

they use a crossover approach which starts simplex iterations and crosses over to

an interior-point method as needed. This is due to the fact that a simplex-type

method (or an active-set approach in nonlinear programming) is quite easy to

restart from a previous solution. In contrast, starting an interior-point method

from the optimal solution of another problem causes issues due to at least one

of a complementary pair of primal-dual variables already being at its bound. A

thorough analysis of the numerical difficulties is presented in [20] and [21] for gen-

eral linear and nonlinear programming warmstarts, respectively, and in [17] and

[18] for warmstarts within branch-and-bound and outer approximation frame-

works, respectively, for mixed-integer nonlinear programming. In all instances, it

is shown that a standard interior-point method, applied directly to the original

problem, will not only fail to warmstart but fatally stall if initialized from the

optimal solution of a previously solved problem.


OPTIMIZATION 105

As pointed out in these papers, the primal-dual penalty approach serves as a

remedy to the stalling issue by un-stalling the iterates and even improves on the

iteration count over a coldstart. This is attained by keeping the optimal values for

the primal-dual variables x, g, y, and y0, but slightly perturbing the primal-dual

relaxation variables ξ and ψ away from 0 (and recomputing f . This perturbation

can be quite small (10−4 usually suffices), since both ξ and ψ are variables and

their values can increase as needed. This framework avoids stalling by moving all

the terms of the complementarity conditions in (3.11) away from 0, but still close

to the central path for a small value of µ.

3.4.5. Handling the discrete variables

For our numerical experiments, we have implemented both a branch-and-bound

method [91] and an outer approximation method [57] for a generic MINLP. Branch-

and-bound conducts a search through a tree where each node is obtained by adding

a bound to its parent to eliminate a noninteger solution and where each node re-

quires the solution of a continuous NLP. Outer approximation alternates between

the solution of an NLP obtained by fixing the integer variables and of an MILP

obtained using linearizations of the objective function and the constraints at the

solutions of the NLP. These methods and their use in conjunction with the primal-

dual penalty interior-point method were analyzed in [17] and [18]. We refer the


OPTIMIZATION 106

reader to these papers for further details.

3.5. Numerical Results

3.5.1. Numerical Results for the Single Period Model

In our numerical testing, we consider one riskless and 20-400 risky assets for trad-

ing. The risky assets are chosen from the S&P500 list of companies in alphabetical

order, and each stock is matched with its real world economic sector. The geomet-

ric mean and the covariance of the risky assets were calculated from the closing

prices of the stocks in 2010. The riskless asset which refers to investment in the

money market has a 1% return.

As we discussed before, we follow both [99] and [29] formulation in our frame-

work. Therefore, we generally use the same constraint parameters with these two

studies for consistency.

Initial weights for the stocks wj = 1/(n+ 1), j = 0, . . . , n

Shortfall risk constraint parameters η1 = 95%, W low1 = 0.90, η2 = 99.7%,

W low2 = 0.95

Diversification by sectors parameters Lmin = ⌈0.5 × L⌉, smin = 0.01

Shortsale portfolio constraints sj = 0.5/n, j = 1, . . . , n, s0 = 0.5


OPTIMIZATION 107

The problem instances are modeled using Matlab and solved using the Matlab-

based solver MILANO ([16]) Version 1.4 which implements both branch-and-

bound and outer approximation algorithms and uses the primal-dual penalty

interior-point approach that allows warmstarting, as described in Section 4. The

mixed-integer LPs arising in the outer approximation algorithm are solved using

Gurobi [117]. Table 1 illustrates the result of the branch and bound algorithm

while Table 2 presents the results of the outer approximation algorithm. The first

column is the number of assets considered for the instance, the second column is

number of different economic sectors, and the third column gives the number of

CVaR constraints included in the model. The next four columns show the num-

bers of nodes and iterations that are required to solve the problem after either a

coldstart or a warmstart. The last column represents the percentage improvement

in the average number of iterations per node, as attained by warmstarting, and

the numbers show that we obtain substantial improvements by using warmstarting

for both the branch-and-bound and outer approximation algorithms.

3.5.2. Numerical Results for the Multi-Period Model

In our numerical testing, we consider 4-10 risky assets for trading. Each asset

is randomly selected from the different economic sectors of S&P500 list of com-


OPTIMIZATION 108

Table 3.1: Results of the Branch-and-Bound Algorithm

Coldstart Warmstart

n L M Nodes Iters Nodes Iters % Impr

20 6 2 7 111 7 63 43.2

50 10 2 25 424 27 282 33.5

100 10 2 33 705 33 446 36.7

200 10 2 11 261 11 184 29.5

400 10 2 19 527 11 238 22.9

Table 3.2: Results of the Outer Approximation Algorithm for Single-Period

Coldstart Warmstart

n L M Nodes Iters Nodes Iters % Impr

20 6 2 2 36 2 31 13.9

50 10 2 3 65 3 52 20.0

100 10 2 3 89 3 60 32.3

200 10 2 2 57 3 69 27.7

400 10 2 2 69 2 53 23.2


OPTIMIZATION 109

panies. Therefore, each asset represents a different, real-world economic sector.

The scenario tree is constructed using monthly returns of the closing price of the

stocks from September 2005 to December 2010.

We use the same constraint parameters as the single period model for consis-

tency.

Initial weights for the stocks wj = 1/(n+ 1), j = 0, . . . , n

Shortfall risk constraint parameters η1 = 95%, W low1 = 0.90, η2 = 99.7%,

W low2 = 0.95

Diversification by sectors parameters Lmin = ⌈0.5 × L⌉, smin = 0.01

Shortsale portfolio constraints sj = 0.5/n, j = 1, . . . , n, s0 = 0.5

The problem instances are modeled and solved as in the single-period case.

Table 3 illustrates the results of the outer approximation algorithm for the multi-

period model. The first column presents the different data sets which are denoted

by TPNS where T represents the number of time period where N represents the

number of stocks in the portfolio. The second column is the number of assets

considered for the instance, the third column is number of different economic

sectors, and the fourth column gives the number of CVaR constraints included

in the model. The next three columns show the number of discrete variables

(DV), the number of continuous variables (CV) and the number of second-order


OPTIMIZATION 110

cone constraint blocks respectively. The next four columns show the numbers of

nodes and iterations that are required to solve the problem after either a coldstart

or a warmstart. The last column represents the percentage improvement in the

average number of iterations per node, as attained by warmstarting, and the

numbers show that we obtain substantial improvements by using warmstarting

for the multi-period model with outer approximation algorithm.


OPTIMIZATION 111

Table 3.3: Results of the Outer Approximation Algorithm for Multi-Period Model

Coldstart Warmstart

Data N L M DV CV SOCC Nodes Iters Nodes Iters % Impr

3P4S 4 4 2 112 168 16 2 45 2 41 8.9

3P6S 6 6 2 168 252 16 2 51 2 46 9.8

3P8S 8 8 2 224 336 16 2 59 2 53 10.2

3P10S 10 10 2 280 420 16 2 60 2 56 6.7

4P4S 4 4 2 240 360 32 2 70 2 66 5.7

4P6S 6 6 2 360 540 32 2 78 2 73 6.4

4P8S 8 8 2 480 720 32 2 82 2 69 15.9

4P10S 10 10 2 600 900 32 2 79 2 69 12.7

5P4S 4 4 2 496 744 64 2 72 2 64 11.1

5P6S 6 6 2 744 1116 64 2 84 2 72 14.3

5P8S 8 8 2 992 1488 64 2 97 2 79 18.6

5P10S 10 10 2 1240 1860 64 5 243 2 89 8.4

CHAPTER 4. REVEALED PREFERENCES FOR PORTFOLIO

SELECTION MODELS 112

Chapter 4

Revealed Preferences for Portfolio

Selection - Does Skewness Matter?

4.1. Introduction

We take a critical look at the paradigm of mean/variance efficient portfolios. Pos-

ing the portfolio selection problem as a decision problem we show how reasonable

assumptions on utility and the probability model lead to asset allocation rules

that depend on mean, variance and skewness of future returns. The main contri-

bution of this article is the empirical validation of this argument by comparing it

with traditional mean variance efficient portfolios. As with any decision problem

the main challenge in setting up a fair comparison of alternative loss functions

is the choice of an appropriate benchmark. Using any of the two competing loss



functions would unfairly bias the comparison in favor of the chosen loss, and the

comparison is meaningless if the other loss function is a better representation of

investor preferences. To enable a meaningful comparison we set up a revealed pref-

erence study. We develop a framework to explain observed investor preferences

by the two alternative utility functions. Minimizing the discrepancy between the

optimal decision under the considered utility functions and the observed data

formalizes the comparison. In other words, we implement the inverse problem

of expected utility maximization. Given observed decisions we back out inference

about the underlying probability model and utility function, revealing the implied

risk preference profile of the investor(s).

We cast the portfolio selection problem as a Bayesian decision problem. The

elements of a decision problem are a probability model for the unknown future

asset returns, a decision variable representing the portfolio choice as a vector of

weights across a given set of assets, and a utility function that models prefer-

ences over consequences. In this context, it can be argued that a rational decision

maker selects a portfolio by maximizing expected utility. The expectation is with

respect to the probability model on the unknown future returns, conditional on

all presently available information, i.e., the posterior predictive distribution. As-

suming that given future returns an investor’s utility is a quadratic function of the

realized returns, it follows that the optimal asset allocation is determined by the

first two moments only. Using a second order expansion, the argument remains



approximately valid for an arbitrary utility function. The apprxomation remains

valid up to a third order expansion if the distribution of future returns follows a

multivariate normal model.

The remainder of this chapter is organized as follows: In Section 4.2. we

describe a probabilty model and a class of utility functions that lead to mean,

variance, skewness efficient portfolios. Section 4.3 develops a framework for a

comparison of alternative utility functions and probability models. In Section

4.4 we report the implementation and results of the proposed comparison with

monthly data from the Dow Jones Industrial Average from August 2008 to January

2013. We will conclude and discuss some future directions of this study in Chapter

6.

4.2. Model

4.2.1. Mean Variance Efficient Portfolios

Markowitz (1952) proposed the idea of selecting portfolio weights based on the cer-

tainty equivalent framework using the mean and variance of historical returns. He

stated that parameter uncertainty should be considered in the allocation problem,

but did not actually address it. In this paper study we follow the implementation

of Harvey et al. (2010) [80] to include parameter uncertainty in the two moment



portfolio problem. This is done by using Bayesian methods complete with drawing

from posterior predictive distributions for the asset returns, and using summaries

of these for estimates for the mean and variance. Specifically we implement the

following setup.

We define the sampling distribution as

(4.1) pm : xtiid∼ N(µ,Σ),

for t = 1 . . . T . The posterior predictive distribution as

pm(xs | x1, . . . , xt), for any future time s > t.

The posterior predictive mean as

mm = E(x | x1, . . . , xt),

where the expectation is with respect to pm, and x = xt+1 generically denotes a

future observation. The utility function is

um(w, x) = w′x− λm[w′(x− mm)]2.

Finally the expected utility to be maximized is

Um(w) = w′mm − λmw′Vmw,

where Vm is the predictive variance.



4.2.2. Asset Allocation with Higher Moments

Harvey et al. (2010) [80] propose a decision problem, i.e., a probability model and

a utility function, to describe portfolio selection with higher order moments. The

probability model is independent sampling from a multivariate skewed normal

distribution. Utility is a third order polynomial of future returns. Details are

described below.

Skew normal distributions provide a technically convenient generalization of

normal models. Several multivariate versions of skew normals have been proposed

in the literature, differing mainly in the number of parameters that are used to

define skewness. Azzalini and Dalla Valle (1996) define a multivariate skew normal

distribution by multiplying a multivariate normal density with a univariate normal

c.d.f. This is generalized by Branco and Dey (2001) and Sahu et al. (2002) by

replacing the univariate normal c.d.f. by a more flexible multivariate normal c.d.f.

We choose the latter to define a probability model for asset returns.

A constructive definition of the multivariate skew normal is as a convolution

of a multivariate normal and a linear function of a truncated multivariate normal.

We say X ∼ SN(µ,Σ,∆) if

(4.2) X = µ+ ∆Z + ǫ with ǫ ∼ N(0,Σ) and Z ∼ N(0, I), Zi ≥ 0.

The distribution is indexed by a location parameter µ, a scale matrix Σ and a

(co-)skewness parameter ∆. Sahu et al. (2002) restrict ∆ to a diagonal matrix. In



Harvey et al. (2010) [80] the model is generalized to unrestricted ∆ to facilitate

inference on co-skewness. Harvey et al. (2010) [80] discuss properties, convenient

prior choices and details of posterior inference for model (4.2).

Let vec(A) denote a representation of an (m × n) matrix A as a (mn × 1)

vector of the stacked columns. We assume a multivariate normal prior for µ and

vec(∆), and a Wishart prior for Σ−1.

We now use the multivariate skew normal distribution to set up a description

of portfolio selection as a decision problem. Let xt denote the returns of the assets

under consideration at time t. We assume

(4.3) ph : xtiid∼ SN(µ,Σ,∆),

t = 1, . . . , T . To simplify notation in the following discussion we generically

use x = xt+1 for future returns in the posterior predictive distribution p(xt+1 |

x1, . . . , xt). Let mh = E(x | x1, . . . , xt) denote the posterior predictive mean. Let

w = (w1, . . . , wp) denote an investor’s portfolio choice, with wi being the relative

weight of the i-th asset. We hypothesize that an investor’s preferences can be

described in terms of future reward w′x and second and third moments:

(4.4) uh(w, x) = w′x− λh[w′(x− m)]2 + γh[w

′(x− m)]3

The function uh(w, x) is the reward for portfolio choice w under assumed future

returns x. The scalars λh and γh are relative weights describing the investor’s



risk averseness. Of course, at the time of the asset allocation decision future re-

turns are unknown. It can be argued that a rational decision maker proceeds by

maximizin expected utility, marginalizing x with respect to the posterior predic-

tive distribituion p(x | x1, . . . , xt). Let Vh and Sh denote the predictive moments.

Then

(4.5) Uh(w) =∫uh dp(x | x1, . . . , xt) = w′m− λhw

′V w + γhw′Sw ⊗ w.

Optimal portfolio selection under the probability model (4.3) and utility (4.4)

proceeds by maximizing U(w) with respect to w.

4.3. Revealed Preferences

We have described two competing descriptions of portfolio selection, the tradi-

tional mean variance efficient portfolio and a generalization allowing for decision

makers to consider skewness in their asset allocation. Both setups are formally

coherent and justifyable as decision theoretically optimal actions. A critical com-

parison of the competing approaches is only possible by validating the models

with observed investor behavior.

We consider a broad based panel of assets, chosen to allow a wide variety of

portfolio choices. At each time t we record total shares outstanding of the stocks

represented in the panel. The relative size wti of the shares outstanding for the

i-th asset in period t quantifies a typical investor’s portfolio weight for asset i. We



refer to wt = (w1, . . . , wp) as the observed portfolio weights. Using wt as observed

data we proceed by finding in each period for both utility functions under consid-

eration the optimal portfolio under the respective utility function that can best

approximate the observed weights wt. We denote with wh∗t and wm∗

t the optimal

portfolio under the higher order moment framework and under the mean variance

efficient framework, respectively. The distances d(wh∗t , wt) and d(wm∗

t , wt) evalu-

ate the fit of the two utility functions to the observed data. Finally, summarizing

the comparison over time provides the desired criterion to evaluate the relative

merit of the two utility functions in the light of the market data. Details are given

in the following algorithm.

In the following description we will use M and H to refer to the two decision

models. Model M refers to the independent normal sampling model (4.1), together

with utility function um(w, x), and Model H refers to skew normal sampling (4.3),

together with utility function uh(w, x).

Algorithm: Revealed Preferences in Asset Allocation

Repeat the follwoing steps 1. through 3. for t = 1, . . . , T

1. Posterior predictive inference.

Find the posterior predictive distributions under both models, pm(x | x1, . . . , xt)

and ph(x | x1, . . . , xt), and evaluate the posterior predictive moments (mm, Vm)



and (mh, Vh, Sh).

2. Find wm∗t and wh∗

t .

2.1. Optimal portfolio for given utility parameters.

Let wm(λm) denote the optimal portfolio under model M, using coeffi-

cient λm.

Let wh(λh, γh) denote the optimal portfolio under model H, using co-

efficients (λh, γh).

2.2. Approximate the observed weights.

Find the utility parameters λm and (λh, γh) that best approximate the

observed data: Let

λmt = arg minλm

d [wt, wm(λm)] .

and similarly for model H:

(λht, γht) = arg minλh,γh

d [wt, wh(λ, γ)] .

2.3. Optimal portfolios to approximate data.

We define wm∗t = wm(λmt) and wh∗

t = wh(λht, γht) as the optimal ap-

proximations to w. Using d(v, w) =∑

(vi −wi)2 the portfolios wh∗

t and

wm∗t are least squares approximations to wh∗

t , under decision models H

and M, respectively.



3. Summarize the approximation residuals.

Plot d(wt, wm∗t ) and d(wt, w

h∗t ) against t. The relative position of the two

curves formalizes the comparison of the two decision models. If desired, a

suitable summary statistic can serve as a single number comparison. For

example, we could use∑t d(wt, w

m∗t ) − d(wt, w

h∗t ).

The described algorithm is highly computation intensive. At each time t we

solve an optimization problem, minimizing the discrepancy between market and

optimal portfolio. The minimization is with respect to the utility parameters λm

and (λh, γh), respectively. Nested within this optimization is a second optimization

problem. For each utility parameter λm (or (λh, γh)) under consideration we solve

another minimization problem to find the optimal portfolio.

4.4. Results

We used daily returns on stock prices for the Dow Jones Industrial Average from

August 2008 to January 2013, a total of 1075 data points as our historical data.

Based on that data we sampled from the posterior predictive distribution for

T = 25 additional steps in to the future.

We were able to show that the three moment decision model uniformly beat

the two moment model in matching the observed portfolio, see Figure 4.1, and

Table 4.1 for comparisons.



Figure 4.1: Distance to the Market Weights: Distance from observed weight to

two and three moment weights. Three moment weights are always closer to the

observed weights.



Table 4.1: Distance to the Market Weights: Distance from observed weight to two

and three moment weights. Three moment weights are always closer to the observed

weights.

Distance to the Market Weights

Data Sets 2 Moments 3 Moments

1076 0.252 0.183

1077 0.260 0.189

1078 0.228 0.189

1079 0.244 0.188

1080 0.244 0.184

1081 0.234 0.196

1082 0.247 0.184

1083 0.235 0.189

1084 0.232 0.185

1085 0.239 0.165

1086 0.249 0.186

1087 0.244 0.185

1088 0.230 0.185

1089 0.230 0.185

1090 0.234 0.189

1091 0.239 0.192

1092 0.232 0.190

1093 0.252 0.188

1094 0.228 0.192

1095 0.247 0.182

1096 0.239 0.187

1097 0.233 0.182

1098 0.246 0.188

1099 0.248 0.174

1100 0.231 0.185



Figure 4.2: The Values of Risk Parameters: The large values for γh suggest that

the typical investor has a strong preference for positive skewness.

Also of interest is the implied risk preferences of the market investor. We can

see from Figure (4.2) and Table (4.2) that λm and (λh, γh) are quite substantial in

magnitude, and quite unstable over time. The rather large values for γh suggest

that the typical investor has a strong preference for positive skewness, which is

consistent with economic theory (see Harvey & Siddique 2000 [81] who argue that

investors are typically willing to trade expected return for positive skewness).



Table 4.2: Risk Parameters

Risk Parameters

2 Moments 3 Moments

Data Sets Lambda Lambda Gamma

1076 1.00E+08 4.80E+05 5.79E+07

1077 2.09E+06 7.54E+05 -7.68E+07

1078 6.58E+05 9.37E+05 7.88E+07

1079 7.68E+07 1.91E+05 2.51E+07

1080 4.03E+06 5.56E+05 -5.85E+07

1081 1.00E+08 4.80E+05 3.96E+07

1082 1.27E+04 4.29E+05 9.01E+07

1083 7.83E+06 4.32E+06 -6.67E+07

1084 6.86E+04 5.77E+05 9.94E+07

1085 4.62E+06 6.02E+05 9.69E+07

1086 7.62E+03 6.02E+05 7.50E+07

1087 8.47E+02 6.23E+07 -5.19E+07

1088 2.12E+06 4.34E+05 -6.53E+07

1089 2.82E+02 4.80E+05 8.87E+07

1090 3.06E+06 2.36E+05 -5.53E+07

1091 3.14E+01 1.44E+06 -9.85E+07

1092 3.14E+01 2.06E+05 7.96E+06

1093 2.46E+07 1.03E+06 -7.46E+07

1094 3.14E+01 1.17E+06 -9.20E+07

1095 1.16E+07 6.02E+05 -5.52E+07

1096 1.23E+06 4.34E+05 8.93E+07

1097 1.84E+06 6.58E+07 -8.37E+07

1098 8.47E+02 2.06E+05 -9.14E+06

1099 7.41E+05 2.04E+05 4.57E+07

1100 8.47E+02 4.80E+05 -3.82E+07



Table 4.3: Current Dow Jones 30 Stocks

Symbol Name

AA Alcoa Inc.

AXP American Express Company

BA The Boeing Company

BAC Bank of America Corporation

CAT Caterpillar Inc.

CSCO Cisco Systems, Inc.

CVX Chevron Corporation

DD E. I. du Pont de Nemours and Company

DIS The Walt Disney Company

GE General Electric Company

HD The Home Depot, Inc.

HPQ Hewlett-Packard Company

IBM International Business Machines Corporation

INTC Intel Corporation

JNJ Johnson & Johnson

JPM JPMorgan Chase & Co.

KO The Coca-Cola Company

MCD McDonald’s Corp.

MMM 3M Company

MRK Merck & Co. Inc.

MSFT Microsoft Corporation

PFE Pfizer Inc.

PG Procter & Gamble Co.

T AT&T, Inc.

TRV The Travelers Companies, Inc.

UNH UnitedHealth Group Incorporated

UTX United Technologies Corp.

VZ Verizon Communications Inc.

WMT Wal-Mart Stores Inc.

XOM Exxon Mobil Corporation

127

Part II

Optimization Problems Arising in


CHAPTER 5. OPTIMIZATION PROBLEMS ARISING IN SUPPLY CHAIN

MANAGEMENT 128

Chapter 5

Multi-Product Batch Production

and Truck Shipment Scheduling

under Different Shipping Policies

5.1. Introduction

Over the past several years, with advances in the notions concerning efficient sup-

ply chains, relationships between customers, manufacturers and suppliers have

undergone numerous notable changes by removing non-value added activities in

procurement, production and distribution. These progressive paradigm changes

tend to view individual decisions as parts of an integrated series of business activ-

ities that span across the entire supply chain. Today’s supply chains are impacted


MANAGEMENT 129

by increased complexity, unpredictable economic conditions, operational risks,

environmental regulations, globalization and rising fuel costs. Historically, opti-

mization projects within the supply chain have been cumbersome, time-consuming

undertakings. Many companies find themselves in a constant struggle to maintain

efficiency at every stage along the supply chain, attempting to reduce costs and

increase productivity within their procurement-production-distribution networks,

in the face of intense competitive pressures. In this context, holistic integra-

tion of decisions involving serial stages of activities has received attention from

researchers in recent years.

This study focuses on a specific supply chain scenario, where a single manu-

facturing plant produces multiple products for satisfying customer demands that

occur at several retail outlets. The production facility can produce only one prod-

uct at a time, but shipments can be made either directly to each individual retailer

via relatively small, less than truckload (LTL) quantities or via larger full truck-

load (TL) quantities, where deliveries are made to all the retailers according to a

peddling arrangement. In the TL transportation mode, a full truckload represents

the aggregate retail demand during a common delivery cycle. The required lot

sizes are then dropped off at the respective retail locations from the same trans-

port vehicle, which incurs a fixed shipping charge. In the case of LTL shipping,

the delivery cycle times for the various products may be different, but any given

item has the same inventory cycle time at all retail locations. The shipments


MANAGEMENT 130

are made directly from the supplier to the various retailers individually and the

respective shipping costs depend on the amount of load delivered, based on a

variable transportation charge.

For either shipment policy, the transportation schedule is directly linked to the

batch production schedule for the multiple items at the manufacturing facility. It

is to be noted that the production batch sizing issue here is represented by the

well-known economic lot scheduling problem (ELSP). Our analysis differs from

existing work in this area in two important ways. First, the inventories of the

different products are depleted at uniform market demand rates in the traditional

treatment of the ELSP, whereas in this paper, we allow such depletions to occur

in discrete lot sizes, depending on the transportation policy in effect. Secondly,

we make an attempt to integrate the production plan with either the TL or LTL

shipment schedule, as the case may be. It is well known that the ELSP addresses

the lot sizing issue for several items with static and deterministic demands over

an infinite planning horizon at a single facility. In this paper, we recast this

problem in a way that ties the production and shipping schedules together with

the objective of minimizing the sum of all the relevant costs, including setup and

other fixed costs, as well as inventory holding and other variable costs, while

satisfying the market demands for all products at the various retail locations.

The solution involves determining a consistent and repetitive production schedule

for all products to meet the necessary demands ([41]). Since the ELSP has been


MANAGEMENT 131

shown to be NP-hard, the focus of most research efforts has been to generate near

optimal cyclic schedules with three well known policies, viz. the common cycle,

basic period (or multiple cycle) and time varying lot size approaches ([133]).

The common cycle (CC) approach always produces a feasible schedule and

is the simplest to implement. However, in some cases, the CC solution, when

compared to the lower bound (LB) solution, turns out to be of poor quality. Unlike

the common cycle approach, the basic period (BP) approach allows different cycle

times for different products, where the individual item cycle times are integer

multiples of a basic period. Although this approach generally tends to yield better

solutions to the ELSP than the CC methodology, obtaining a feasible schedule is

NP-hard ([27]). Moreover, the computational effort required for implementing

the BP solution is considerable greater compared to the CC solution. Finally,

the time-varying lot size approach, being more ï¬‚exible than the aforementioned

procedures, allows for different lot sizes for the different products in a cycle. In

[52], Dobson showed that the time-varying lot size technique always produces

feasible schedules, while generating better quality solutions. Nevertheless, the

computational burden associated with this procedure is significantly higher than

those for the other two approaches. Thus, in order to keep the computational

complexity to a minimum, as well for the sake of simplicity of implementation, we

adopt the common cycle approach in our integrated analysis for addressing the

production batching issue.


MANAGEMENT 132

This chapter attempts to extend the classical ELSP model by incorporating

the transportation decision, accounting for finished goods inventories in discrete,

sizeable lots. It may be beneficial to deliver quantities of the various products,

using either the full truckload (TL) or less than truckload (LTL) shipment policies.

In the case of the TL policy, each truckload consists of a mix of all the items. As

mentioned earlier, we also adopt, for simplicity, the common production cycle

(CC) approach (see, for example, [105]), with a delivery cycle that is common

to all the individual items. For coordination purposes, this delivery cycle is a

multiple integer of the overall production cycle, also common to all items. Under

the LTL shipping policy, the different items may have different delivery cycles,

where individual products are shipped directly to the retailers. Nevertheless,

for each product, each item’s delivery cycle is an integer multiple of the overall

production cycle, which is common to all the products.

This work extends the multiproduct model presented by Banerjee [8]. He for-

mulated an analytical model to align the production schedule of multiple products

with a full truckload delivery plan and develops a heuristic solution methodology.

We propose a generalized mixed integer, non-linear mathematical programming

model (MINLP) for developing a multi-product batch production schedule, which

coordinates finished goods availabilities with their outbound TL or LTL shipment

plans. The transportation cost for a TL shipment is a fixed cost, whereas LTL

shipment costs are based on a variable shipping charge. Finally, the models and


MANAGEMENT 133

the concepts regarding coordinated production and shipment decisions developed

in this study are illustrated through numerical examples.

The remainder of this chapter is organized as follows: The next section of this

chapter outlines the assumptions made and the notation used in our models and

in Section 3, the proposed models are described in detail. A numerical example

and some selected sensitivity analyses are presented in Section 4. Finally, Section

5 provides a summary and some concluding remarks.

5.2. Assumptions and notation

In this Section we present the assumptions that we need and the important nota-

tion that we use throughout this paper.

5.2.1. Assumptions

The following assumptions are made in describing the manufacturing-distribution

scenario adopted in this paper and for formulating our models that follow:

1. Market demands for the various products are deterministic and stationary.

2. A set of products are manufactured in a single capacitated batch production

facility, with different production rates for the various items.

3. Only a single product may be produced at any given time.


MANAGEMENT 134

4. Stockouts are not permitted.

5. The common cycle (CC) approach is deployed to solve the ELSP, where each

product is produced exactly once in every production cycle.

6. Each of the products is transported via truck and is delivered to one or more

given demand locations, depending on one of two shipping policies in effect.

7. Under a full truckload (TL) shipping policy, a mix of all products, constitut-

ing a full load, is delivered to all retail locations on the basis of a peddling

arrangement. The LTL transportation mode, on the other hand, implies

direct shipment of each product to each retailer.

8. These two scenarios impose different transportaion costs. The TL mode

involves a capacitated vehicle, incurring only a fixed cost for all the peddling

shipments made in a single delivery, while for LTL shipments, each direct

shipment cost is based on a load-based variable cost

9. Under the TL policy, an integer number, K, of deliveries are made at equal

intervals of time over a production cycle.

10. For the LTL case, the number of deliveries made per common production

cycle may vary for the different products, but are still integer multiples,

K1, K2, etc., of the production cycle.


MANAGEMENT 135

11. At each of the various demand locations, stocks are replenished via a periodic

review, order-up-to level inventory control system, when TL shipping is in

effect. For coordination purposes, all the items at all the demand locations

share a common fixed review period.

12. Under the LTL shipment policy, although the review periods for the various

items may be different, for any given product, all retail locations share a

common review period, for coordination purposes.

5.2.2. Notation

The notational scheme is adopted in the formulation of our models is given in

Table 5.1.

5.3. Model Development

In this section, we present the details of the two shipment policies adopted in this

paper, based on direct shipment and peddling shipment modes. These distribution

policies are depicted in Figure 5.1.


MANAGEMENT 136

Table 5.1: Notation

i An index used to denote a specific product, i = 1, 2, . . . , n

Di The demand rate for product i (units/time unit)

Pi The production rate for product i (units/time unit)

Ai Manufacturing setup cost per production batch for product i ($/batch)

hi Inventory holding (carrying) cost for product i ($/unit/time unit)

Qi Amount of product i contained in each TL shipment (units)

K A positive integer, representing the number of shipments per production cycle

T The shipment interval in time units (common to all products and locations)

KQi The production lot size (in units) for product i

KT Production cycle length in time units

C The FTL capacity, i.e. maximum total load (or volume) allowable per truckload

wi Weight (or volume) of each unit of product i

It Average inventory level (units) of product i

TRC Total relevant cost ($) per time unit

γ Fixed cost of initiating one truck dispatch ($/dispatch) for TL policy

v Unit shipment cost of products for less than truckload (LTL) amounts ($/pound).


MANAGEMENT 137

Figure 5.1: Direct Shipping (LTL) vs. Peddling Shipping (TL) Policies

5.3.1. Shipment Policies

Our analyses are based on extensions of the multiproduct model presented in

Banerjee (2009). We develop an analytical model and methods to minimize total

inventory and transportation related costs when a supplier distributes a set of dif-

ferent items to several retailers or customers. This paper evaluates and compares

two different distribution policies: direct shipping and peddling.

The direct shipping distribution policy involves shipping separate loads from

the supplier directly to each customer, whereas peddling shipping dispatches a

fully loaded truck in each distribution cycle, that deliver items to all of the cus-

tomers, based on each locations demand during this cycle. The latter is depicted

in Figure 5.2 and a LTL distribution situation is shown in Figure 5.3.


MANAGEMENT 138

5.3.2. TL policy model formulation

For illustrative purposes, the inventory-time plots for a TL distribution scenario

are shown in Figure 5.2. This plot illustrates a situation that involves three

products (n = 3) with negligibly small set up times and transit times and three

full TL shipments for each production cycle. Figure 5.2 shows that there is a

common delivery cycle time of T . Each truck with a limited capacity, C, contains

Qi units of product i, (i = 1, 2, 3). The products should be sequenced to minimize

the total set up cost, inventory holding cost and transportation cost (see [8], for

an explanation of this).

We obtain the following the average inventory values for the three items:

I1 =[ 1

2KQ1( KQ1

P1) +KQ1( KQ2

P2+ Q3

P3) + (K − 1)( Q1

D1)Q1 + (K − 2)( Q1

D1)Q1 + · · · + ( Q1

D1)Q1]

KQ1/D1

= KD1(Q1

2P1

+Q2

P2

) +D1(Q2

P2

) + (K − 1)(Q1

2)

I2 =[ 1

2KQ2( KQ2

P2) +KQ2( Q3

P3) + (K − 1)( Q2

D2)Q2 + (K − 2)( Q2

D2)Q2 + · · · + ( Q2

D2)Q2]

KQ2/D2

= KD2(Q2

2P2

) +D2(Q3

P3

) + (K − 1)(Q2

2)

I3 =Q2

2[D3

P3

(2 −K) + (K − 1)]

5.3.2.1. Objective Function

In consideration of these results, we formulate the minimization objective function

(the total relevant cost per time unit), as shown below. This expression includes

the inventory holding, setup, and the transportation costs per time unit. Note

that the cost function below is non-linear, with an integrality requirement. The


MANAGEMENT 139

Figure 5.2: Inventory-Time Plots for a Peddling Shipment Policy (n = 3, K = 3)

(As shown in [8])


MANAGEMENT 140

decision variables are the amounts of all the products, Qi (for all i), contained

in each TL shipment and the number of shipments per production cycle which is

denoted by K, an integer.

Minimize TRC(Q,K) =

n∑

i=1

DiAi

KQi+K

n−1∑

i=1

Dihi[Qi

2Pi

+

n−1∑

j=i+1

Qj

Pj

] +Qn

Pn

n−1∑

i=1

Dihi+

(K − 1)Qn

Pn

n−1∑

i=1

Qihi

2+Qnhn

2[Dn

Pn

(2 −K) + (K − 1)] + γ1

T

The first term above represents the total manufacturing setup cost per production

batch for n products. The last part of the objective function represents the total

transportation cost that is obtained by the multiplication of the fixed cost of

initiating one truck dispatch and the total number of TL shipment per unit of

time. The remaining terms capture the inventory holding cost per time unit

for items 1, 2, . . . , n, respectively. Finally, we obtained convex objective function

which is shown in Appendix.

5.3.2.2. Constraints

1. The delivery cycle is common to all products:

Q1

D1=Q2

D2= · · · =

Qn

Dn= T or Qi = TDi where i = 1, 2, . . . , n

2. Production schedule should be feasible, i.e. total production time should

be less than the manufacturing cycle time (without loss of generality, we


MANAGEMENT 141

assume that the manufacturing setup times are negligibly small):

Kn−1∑

i=1

Qi

Pi+Qn

Pn≤ KT or

n−1∑

i=1

Qi

Pi+

Qn

KPn≤ T

3. The load capacity of a truck is limited by the total weight (or volume) of

the products, i.e.n∑

i=1

wiQi = C

4. At least one TL shipment must be made over a production cycle:

K ≥ 1 where K ∈ Z+

The TL policy, i.e. a mixed integer non-linear programming (MINLP), model for-

mulated above may be solved using one of several computer based solvers available.

We employ the BONMIN solver for this purpose and obtain the optimal solution.

5.3.3. LTL policy model formulation

For illustrative purposes, the inventory-time plots for a direct shipment based LTL

distribution policy are shown in Figure 5.3, which represemts a scenario involving

three products (n = 3) with negligible set up and transit times. Note that LTL

shipment for each production cycle.


MANAGEMENT 142

Figure 5.3: Inventory-Time Plots for a Direct Shipment Policy (n = 3, K1 =

4, K2 = 3, K3 = 1)


MANAGEMENT 143

5.3.3.1. Objective Function

As before, the objective is to minimize the total relevant cost per time unit,

which includes the inventory holding, setup and the transportation costs. Also,

the decision variables consist of the amount of product i contained in each LTL

shipment (denoted by Qi) and the number of shipments per production cycle for

product i, Ki, which are restricted to positive integers. The objective function of

the LTL model then can be expressed as:

Minimize TRC(Q,K) =1

τ

n∑

i=1

Ai + τn∑

i=1

Dihi2

[Di

Pi(2 − 1

Ki)] +

n∑

i=1

wiDivi

where τ = KiTi.

The first term above represents the total manufacturing setup cost per production

batch for n products, second term captures the total inventory holding cost for

all items and the last term denotes the total transportation cost per unit of time.

Finally, we obtained convex objective function which is shown in Appendix.

5.3.3.2. Constraints

1. The delivery cycle time is common to all products:

Q1

D1

=Q2

D2

= · · · =Qn

Dn

= τ ; τ = KiTi so Qi = DiKiTi, i = 1, 2, . . . , n

2. At least one shipment per truck should be made within a production cycle:

K ≥ 1∀i, where K ∈ Z+


MANAGEMENT 144

Table 5.2: Example Problem Parameters

Product Di Pi Ai hi wi

(i) (units/year) (units/year) ($/setup) ($/unit/year) (lbs./unit)

1 8,000 30,000 1,500 40 20

2 12,000 50,000 3,000 72 50

3 15,000 40,000 2,400 60 40

Once again, the BONMIN solver is utilized so find the optimal solution to the

MINLP LTL policy model formulated above.

5.4. Numerical Example

This section presents an illustrative example involving three products. The rele-

vant data pertaining to the problem are shown in Table 5.2.

Truck capacity is varied between 10000 lbs. and 70000 lbs. at 5000 lbs.

increments, for full truckload shipments. In addition, the unit variable shipment

cost of products for the less than truckload mode is varied from $0.24 to $0.18 per

lb. in increments of $0.01. As mentioned before, we obtain the optimal solutions

to the mixed integer nonlinear optimization problems (MINLPs) for both TL and


MANAGEMENT 145

LTL shipment policies using the BONMIN solver, which provides global optimal

solutions for the MINLPs. Table 5.3 presents the summary of the computational

results for TL shipments with varying truck capacities and Table 5.4 shows the

summary results for the LTL shipping policy, incorporating different unit variable

shipping costs.

Table 5.3 indicates the TL delivery cycle time varies from 0.00735 year to

0.05147 year. These increase as the truck capacity goes up. The number of TL

deliveries per production cycle (K ) tends to decrease with increasing truck capac-

ity. These results are not unexpected, since the fixed cost per shipment tends to

increase with larger vehicle capacities. To compensate for this phenomenon, the

production cycle time is increased, together with fewer deliveries per manufactur-

ing cycle. Needless to say that delivery lot sizes also increase with higher truck

capacities. Interestingly, the total relevant cost function value tends to exhibit

a convex behavior with respect to vehicle capacity. Clearly, due to the effects of

economies of scale, the TRC decreases with a larger and larger vehicle size. Nev-

ertheless, after a certain truck size, the initial cost advantage of scale seems to be

more than offset by the higher annual truck dispatching costs, as well as higher

inventory holding costs, resulting from the need to hold more output in stock


MANAGEMENT 146

Table 5.3: Numerical Results for TL Shipment Policy with Varying Truck Capac-

ities

Capacity Gamma T K Q1 Q2 Q3 TRC

10000 3000 0.00735294 10 58.82 88.24 110.29 595789

15000 3300 0.01102940 7 88.24 132.35 165.44 486356

20000 3630 0.01470590 5 117.65 176.47 220.59 432714

25000 3993 0.01838240 4 147.06 220.59 275.74 402135

30000 4392 0.02205880 3 176.47 264.71 330.88 383902

35000 4832 0.02573530 3 205.88 308.82 386.03 371083

40000 5315 0.02941180 3 235.29 352.94 441.18 366286

45000 5846 0.03308820 2 264.71 397.06 496.32 358604

50000 6431 0.03676470 2 294.12 441.18 551.47 355050

55000 7074 0.04044120 2 323.53 485.29 606.62 355146

60000 7781 0.04411760 2 352.94 529.41 661.77 358114

65000 8559 0.04779410 2 382.35 573.53 716.91 363439

70000 9415 0.05147060 1 411.77 617.65 772.06 370674


MANAGEMENT 147

Table 5.4: Numerical Results for LTL Shipment Policy

T Ki Q1 Q2 Q3 TRC v

0.14798 1,1,1 1183.84 1775.76 2219.7 419656 0.24

0.14798 1,1,1 1183.84 1775.76 2219.7 406056 0.23

0.14798 1,1,1 1183.84 1775.76 2219.7 392456 0.22

0.14798 1,1,1 1183.84 1775.76 2219.7 378856 0.21

0.14798 1,1,1 1183.84 1775.76 2219.7 365256 0.20

0.14798 1,1,1 1183.84 1775.76 2219.7 351656 0.19

0.14798 1,1,1 1183.84 1775.76 2219.7 338056 0.18

before a larger vehicle can be fully loaded. For the given problem parameters, it

appears that under a peddling distribution policy, TL shipments with a 50,000

lbs. truck capacity yields the lowest total relevant cost per year of $355,050.

The results for the LTL direct shipment policy, as shown in Table 5.4, lead

to some interesting observations. First, in the absence of a fixed shipping cost, a

common production cycle leads to a lot-for-lot (with respect to aggregate market

demand) delivery policy for each of the products concerned, i.e. Ki = 0, ∀i. From

the minimization objective function of the LTL policy model, it is clear that for

any given production cycle time, τ , the second term, representing the total holding

cost, is minimal when all Ki values are set to 0. Thus, if a feasible solution (i.e.


MANAGEMENT 148

Figure 5.4: Comparison of TL and LTL Shipment Policies

sufficient production capacity) exists for this model, the optimization task in this

case is to determine the appropriate value of τ . Once this is accomplished, the

problem is essentially solved. Hence, we observe that regardless of the value of

the unit variable shipping charge, the production and delivery cycles remain the

same, although, the annual total relevant cost increases with increasing variable

transportation cost. For the example chosen, the optimal cycle time remains fixed

at 0.14798 year, with the same lot-for-lot product deliveries for differing variable

shipping charges. Figure 5.4, comparing the TRC values for the TL policies with

varying truck capacities and the

LTL policy with changing variable shipping costs, further indicates that when

the unit shipping cost is sufficiently low, the latter policy is always superior from

a cost perspective. Otherwise, there is clearly a beak-even point between these

two policies, with respect to vehicle capacity. For relatively small-sized trucks, the


MANAGEMENT 149

Figure 5.5: Policy Comparison with 250% Increase in Setup Cost

LTL policy is likely to be more desirable, whereas the TL shipping policy tends

to yield lower TRC values, beyond the break-even truck capacity level, before the

effect of diseconomies of scale take effect. This is not surprising, since with larger

trucks the fixed charge structure is based on a decreasing prorated cost per unit

shipped.

For the purpose of sensitivity analysis, we vary the manufacturing setup cost,

Ai, and the fixed TL shipping cost, γ, values. The results of these analysis are

summarized in Figures 5.5, 5.6, 5.7 and 5.8, which

indicate that with increasing fixed TL shipping charges, the LTL policy tends

to become more dominant. By the same token, if this cost decreases, the TL

policy tends to be superior to the LTL shipping mode. Additionally, increasing

the production setup cost appears to have a similar effect with respect to the two

distribution policies examined here. In other words, all else being equal, higher


MANAGEMENT 150

Figure 5.6: Policy Comparison with 500% Increase in Setup Cost

Figure 5.7: Policy Comparison with 50% Reductions in TL Shipping Costs


MANAGEMENT 151

Figure 5.8: Policy Comparison with 100% Increase TL Shipping Costs

setup costs tend to render the LTL policy a better alternative to TL distribution.

CHAPTER 6. CONCLUSIONS AND FUTURE RESEARCH 152

Chapter 6

Conclusions and Future Research

Directions

6.1. Conclusions

In this chapter, we will provide concluding remarks for each of the applications

covered in Chapters 3, 4 and 5. The overall contribution of this dissertation is

the use of these applications to motivate the development and use of advanced

statistical and optimization techniques to solve business problems. Given the

success of our solution methods and/or existing approaches on these applications,

we believe that we have provided sufficient motivation for future researchers.


6.1.1. Portfolio Selection Models as MISOCPs

In Chapter 2, we gave an overview of the state-of-the-art in mixed-integer second-

order cone programming problems. We described numerous applications and a

handful of solution algorithms. Given the wide range of fields from which the

applications arise, we anticipate that this problem class will continue to flourish.

The solution methods for MISOCP are still at their infancies, however, so for

the growth of this problem class, it is important to continue to address issues

of warmstarts and levels of accuracy in methods for solving the continuous re-

laxations and to add to the types of cuts available to improve the efficiency of

overall solution approaches. The lifted LP branch-and-bound algorithm presents

another opportunity for algorithmic improvement, and it may be useful to investi-

gate other approaches for solving SOCPs using an LP-based approach within the

MISOCP framework.

In Chapter 3, we presented a set of techniques for solving MISOCPs as MINLPs

whose underlying NLPs are smooth, regularized, and convex. A ratio reformula-

tion was used to smooth the underlying SOCPs. The primal-dual penalty interior-

point method, modified from that presented in [17] and [18], was then used to

provide warmstarts, regularization, and infeasibility detection capabilities, and

the modification also exploited the structure of the MISOCP. We have imple-

mented both branch-and-bound and outer approximation frameworks that use


this method, and use them to solve portfolio optimization problems. Numerical

results show that we can solve small to medium-sized instances successfully. The

infeasibility detection capability provided by the primal-dual penalty approach

allows us to either solve or declare infeasibility at each node, thereby leading to

a robust method. The warmstart capability is shown to significantly improve

algorithm efficiency.

In future work, we hope to extend our approach to general MISOCPs by having

a dynamic choice of constraint reformulations to resolve nonsmoothness issues.

For handling the integer variables, our proposed frameworks can accommodate

the various cuts appearing in MISOCP literature, and we will investigate such

algorithmic improvements as well. Additionally, we will continue our work on

portfolio optimization models by working to include round-lot constraints in our

models for both single and multi-period portfolio optimization model.

6.1.2. Skewness in Portfolio Selection Models

We have proposed and implemented a competition between traditional mean vari-

ance efficient portfolio selection and an alternative portfolio selection paradigm

based on higher order moments. We have shown that the higher order moment

model does a better job of describing the "typical investor’s" portfolio and allows

us to estimate the revealed preferences of the market.


The comparison is fair in the sense that it is based on market data and is

not unfairly hinged upon one or the other decision criterion. However, several

limitations remain. Perhaps the most important limitations are related to the

appropriate interpretation of the market data. We used total shares outstanding

of stocks in a broad-based set of assets to define maket weights that reflect a

“typical investor” and proceeded to approximate these weights under the two

models of interest. But of course the sum of the optimal solutions of all investors

does not necessarily take the form of the optimal solution of an average investor.

Another limitation is the constraint to a fixed set of assets. In reality, investors

have choices beyond the limited number of assets considered. We mitigate this

problem by considering a widely diversified mix of assets.

6.1.3. Supply Chain Management

In Chapter 5, we have made an attempt to integrate the lot scheduling decisions

for multiple products produced in a single facility, with their shipment schedules

under two different types of transportation cost structures under deterministic

conditions. One common type of shipping rate regime found in the real world

involves full truckload (TL) or carload movement of goods, where only a fixed

cost is incurred depending on the points of origin and destination, as well as the

type of commodity moved. An alternative transportation mode is the less than


truckload (LTL), or carload shipping, where there is no fixed cost. The cost of a

specific shipment is based on a variable cost per unit moved from an origin to a

destination. In our analysis, we incorporate both of these transportation scenarios

for a single manufacturer and several retailers. Furthermore, under a TL shipping

policy, we employ a peddling type of distribution arrangement, where a fully

loaded vehicle containing a mix of all the products is dispatched to all the retail

locations for simultaneous delivery. In the case of LTL shipments, each product

has its own delivery cycle and shipments are made directly and individually to

each of the retailers when a batch of the item is completed.

We construct constrained (MINLP) models for linking the production and dis-

tribution decisions under both of the distribution policies described above and

employ widely available solver software for finding globally optimal solutions.

Through a set of numerical experiments we show that the respective magnitudes

of the various cost parameters play a crucial role in selecting either a TL or

LTL distribution method. An important finding of this work is that when trans-

portation involves no fixed cost, but only a variable charge per unit shipped, the

optimal shipment schedule is essentially lot-for-lot with respect to aggregate retail

demand. We have observed that under TL distribution, the production, as well as

the delivery cycle lengths tend to go as vehicles of larger capacities are employed.

Also, we have attempted to outline the parametric conditions under which either

of the two transportation modes will dominate the other from a cost perspective.


It is hoped that this study will provide a helpful tool for supply chain practi-

tioners, in terms of integrating the production schedule with transportation plan-

ning and selecting an appropriate method of distribution. We also hope that

future research endeavors in this area will find some value in this work and will

extend our findings under more complex and realistic supply chain environments.

6.2. Future Research Directions

As we mentioned before, mixed-integer second-order cone programming problems

arise a variety of important application areas ranging from finance to electrical

engineering, from operations management to statistics because of two types of

constraints: (1) risk (volatility) constraints can easily be formulated as second-

order cone constraints, and (2) binary choices and discrete decisions are quite

common in the real world. Therefore, exploring the use of the improved solution

approaches proposed here to a variety of different application areas will be a

significant part of my future research agenda. Given the current advanced state

of my research, I believe that I am poised to make contributions to the various

fields in a timely manner.

In fact, we have already started working on the solution of a humanitarian

logistics problem that arises in a real-world application. ABC is a utility company

that delivers natural gas to its customers via underground pipelines. This utility


company conducts the installation and maintenance of the pipelines as well as

responds to gas leak emergencies. It wants to improve the operational performance

of its emergency crews which one subject to response to the constraints by state

law. We are using real-world historical data which was gathered the dispatching

office of the company from the gas leakage calls. The company’s objective is to

find the optimal number of centers that minimize the total travel distance from

employees’ homes to centers and from centers to leakage areas subject to the state-

law constraint. Second-order cone constraint arise in the calculation of the total

travel distance and binary variables arise in assigning each employee to centers

and centers to the leakage areas. Therefore, the model is formulated as a k-centers

problem that fits the MISOCP framework. Another version of the problem uses

nonlinear constraints to formulate distances using latitude-longitude information,

resulting in the overall k-centers problem to be formulated as a MINLP.

The techniques and applications discussed in this dissertation readily extend

to problems such as that of ABC. I look forward to making further contributions

to the field by pursuing research in such directions.

BIBLIOGRAPHY 159

Bibliography

[1] CJ Adcock and N. Meade. A simple algorithm to incorporate transactions

costs in quadratic optimisation. European Journal of Operational Research,

79(1):85–94, 1994.

[2] F. Alizadeh and D. Goldfarb. Second-order cone programming. Mathemat-

ical Programming, 95:3–51, 2003.

[3] Farid Alizadeh, Jean-Pierre A Haeberly, and Michael L Overton. Primal-

dual interior-point methods for semidefinite programming: convergence

rates, stability and numerical results. SIAM Journal on Optimization,

8(3):746–768, 1998.

[4] A. Atamturk, Gemma Berenguer, and Zou-Jun Max Shen. A conic inte-

ger programming approach to stochastic joint location-inventory problems.

Operations Research, 60:366–381, 2012.

[5] A. Atamturk and V. Narayanan. Cuts for conic mixed-integer programming.

BIBLIOGRAPHY 160

In Proceedings of IPCO 2007, M. Fischetti and D.P. Williamson, eds., pages

16–29, 2007.

[6] A. Atamturk and V. Narayanan. Conic mixed-integer rounding cuts. Math-

ematical Programming Series A, 122:1–20, 2010.

[7] E. Balas, S. Ceria, and G. Cornuejols. A lift-and-project cutting plane

algorithm for mixed 0-1 programs. Mathematical Programming, 58:295–324,

1993.

[8] Avijit Banerjee. Simultaneous determination of multiproduct batch and full

truckload shipment schedules. International Journal of Production Eco-

nomics, 118(1):111–117, 2009.

[9] Mesut E Baran and Felix F Wu. Network reconfiguration in distribution

systems for loss reduction and load balancing. Power Delivery, IEEE Trans-

actions on, 4(2):1401–1407, 1989.

[10] Jonathan F Bard. Coordination of a multidivisional organization through

two levels of management. Omega, 11(5):457–468, 1983.

[11] Omar Ben-Ayed, David E Boyce, and Charles E Blair III. A general bilevel

linear programming formulation of the network design problem. Transporta-

tion Research Part B: Methodological, 22(4):311–318, 1988.

BIBLIOGRAPHY 161

[12] A. Ben-Tal and A. Nemirovski. On polyhedral approximations of the second-

order cone. Mathematics of Operations Research, 26(2):193–205, May 2001.

[13] Julian Benjamin. An analysis of inventory and transportation costs in a

constrained network. Transportation Science, 23(3):177–183, 1989.

[14] Hande Y Benson and Ümit Saglam. Mixed-integer second-order cone pro-

gramming: A survey. In Tutorials in Operations Research, pages 13–36.

INFORMS, 2013.

[15] Hande Y Benson and Ümit Saglam. Smoothing and regularization for mixed-

integer second-order cone programming with applications in portfolio opti-

mization. In Modeling and Optimization: Theory and Applications, pages

87–111. Springer, 2013.

[16] H.Y. Benson. MILANO - a Matlab-based code for mixed-integer linear and

nonlinear optimization. http://www.pages.drexel.edu/˜hvb22/milano.

[17] H.Y. Benson. Mixed-integer nonlinear programming using interior-point

methods. Optimization Methods and Software, 26(6):911–931, 2011.

[18] H.Y. Benson. Using interior-point methods within an outer approximation

framework for mixed integer nonlinear programming. In Jon Lee and Sven

Leyffer, editors, Mixed Integer Nonlinear Programming, volume 154 of IMA

BIBLIOGRAPHY 162

Volumes on Mathematics and Its Applications, pages 225–243. Springer New

York, 2012.

[19] H.Y. Benson, A. Sen, and D.F. Shanno. Convergence analysis of an interior-

point method for nonconvex nonlinear programming. Technical report,

February 2009.

[20] H.Y. Benson and D.F. Shanno. An exact primal-dual penalty method

approach to warmstarting interior-point methods for linear programming.

Computational Optimization and Applications, 38(3):371–399, December

2007.

[21] H.Y. Benson and D.F. Shanno. Interior-point methods for nonconvex non-

linear programming: Regularization and warmstarts. Computational Opti-

mization and Applications, 40(2):143–189, June 2008.

[22] H.Y. Benson, D.F. Shanno, and R.J. Vanderbei. A comparative study of

large scale nonlinear optimization algorithms. In Proceedings of the Work-

shop on High Performance Algorithms and Software for Nonlinear Opti-

mization, Erice, Italy, 2001.

[23] H.Y. Benson and R.J. Vanderbei. Solving problems with semidefinite and

related constraints using interior-point methods for nonlinear programming.

Mathematical Programming B, 95(2):279–302, 2003.

BIBLIOGRAPHY 163

[24] D. Bertsimas and D. Pachamanova. Robust multiperiod portfolio manage-

ment in the presence of transaction costs. Computers & Operations Research,

35(1):3–17, 2008.

[25] Rohit Bhatnagar, Pankaj Chandra, and Suresh K Goyal. Models for multi-

plant coordination. European Journal of Operational Research, 67(2):141–

160, 1993.

[26] Dennis E. Blumenfeld, Lawrence D. Burns, J.David Diltz, and Carlos F.

Daganzo. Analyzing trade-offs between transportation, inventory and pro-

duction costs on freight networks. Transportation Research Part B: Method-

ological, 19(5):361 – 380, 1985.

[27] Earl E Bomberger. A dynamic programming approach to a lot size schedul-

ing problem. Management Science, 12(11):778–784, 1966.

[28] P. Bonami, L.T. Biegler, A.R. Conn, G. Cornuejols, I.E. Grossman, C.D.

Laird, J. Lee, A. Lodi, F. Margot, N. Sawaya, and A. Waechter. An algo-

rithmic framework for convex mixed integer nonlinear programs. Discrete

Optimization, 5:186–204, 2008.

[29] P. Bonami and M.A. Lejeune. An exact solution approach for portfolio

optimization problems under stochastic and integer constraints. Operations

research, 57(3):650–670, 2009.

BIBLIOGRAPHY 164

[30] Jerome Bracken and James T McGill. Mathematical programs with op-

timization problems in the constraints. Operations Research, 21(1):37–44,

1973.

[31] R. Brandenberg and L. Roth. New algorithms for k-center and extensions.

In Proceedings of COCOA 2008, B. Yang, D-Z. Du, and C.A. Wang, eds.,

pages 64–78. Springer-Verlag, 2008.

[32] D.B. Brown and J.E. Smith. Dynamic portfolio optimization with transac-

tion costs: Heuristics and dual bounds. Management Science, 57(10):1752–

1770, 2011.

[33] Anthony P Burgard, Priti Pharkya, and Costas D Maranas. Optknock: a

bilevel programming framework for identifying gene knockout strategies for

microbial strain optimization. Biotechnology and bioengineering, 84(6):647–

657, 2003.

[34] G.C. Calafiore. Multi-period portfolio optimization with linear control poli-

cies. Automatica, 44(10):2463–2473, 2008.

[35] Wilfred Candler, Jose Fortuny-Amat, and Bruce McCarl. The potential role

of multilevel programming in agricultural economics. American Journal of

Agricultural Economics, 63(3):521–531, 1981.

BIBLIOGRAPHY 165

[36] Wilfred Candler and Roger D Norton. Multi-level programming. Number 20.

World Bank, 1977.

[37] RG Cassidy, MJL Kirby, and WM Raike. Efficient distribution of resources

through three levels of government. Management Science, 17(8):B–462,

1971.

[38] U. Celikyurt and S. Özekici. Multiperiod portfolio optimization models in

stochastic markets using the mean–variance approach. European Journal of

Operational Research, 179(1):186–202, 2007.

[39] M.T. Cezik and G. Iyengar. Cuts for mixed 0-1 conic programming. Math-

ematical Programming Series A, 104:179–202, 2005.

[40] Pankaj Chandra and Marshall L Fisher. Coordination of production and

distribution planning. European Journal of Operational Research, 72(3):503–

517, 1994.

[41] Dean C Chatfield. The economic lot scheduling problem: A pure genetic

search approach. Computers & Operations Research, 34(10):2865–2881,

2007.

[42] Y. Cheng, S. Drewes, A. Philipp, and M. Pesavento. Joint network opti-

mization and beamforming for coordinated multi-point transmission using

BIBLIOGRAPHY 166

mixed integer programming. In Proceedings of ICASSP 2012, pages 3217–

3220, 2012.

[43] Suh-Wen Chiou. Bilevel programming for the continuous transport network

design problem. Transportation Research Part B: Methodological, 39(4):361–

383, 2005.

[44] Peter A Clark and Arthur W Westerberg. Bilevel programming for steady-

state chemical process design-I. fundamentals and algorithms. Computers

& Chemical Engineering, 14(1):87–97, 1990.

[45] Benoît Colson, Patrice Marcotte, and Gilles Savard. An overview of bilevel

optimization. Annals of Operations Research, 153(1):235–256, 2007.

[46] Andrew R Conn and Luís N Vicente. Bilevel derivative-free optimization and

its application to robust optimization. Optimization Methods and Software,

27(3):561–577, 2012.

[47] Jean-Philippe Côté, Patrice Marcotte, and Gilles Savard. A bilevel mod-

elling approach to pricing and fare optimisation in the airline industry. Jour-

nal of Revenue and Pricing Management, 2(1):23–36, 2003.

[48] George B Dantzig and Gerd Infanger. Multi-stage stochastic linear pro-

grams for portfolio optimization. Annals of Operations Research, 45(1):59–

76, 1993.

BIBLIOGRAPHY 167

[49] Frans A De Roon, Theo E Nijman, and Bas JM Werker. Testing for mean-

variance spanning with short sales constraints and transaction costs: The

case of emerging markets. The Journal of Finance, 56(2):721–742, 2002.

[50] Christian M Delporte and L Joseph Thomas. Lot sizing and sequencing for

n products on one facility. Management Science, 23(10):1070–1079, 1977.

[51] Stephan Dempe. Foundations of bilevel programming. Springer, 2002.

[52] Gregory Dobson. The economic lot-scheduling problem: achieving feasibility

using time-varying lot sizes. Operations Research, 35(5):764–771, 1987.

[53] C Loren Doll and D Clay Whybark. An iterative procedure for the

single-machine multi-product lot scheduling problem. Management Science,

20(1):50–55, 1973.

[54] S. Drewes. Mixed integer second order cone programming. PhD thesis.

Technischen Universitat Darmstadt, Munich, Germany., 2009.

[55] Y. Du, Q. Chen, X. Quan, L. Long, and R.Y.K. Fung. Berth allocation

considering fuel consumption and vessel emissions. Transportation Research

Part E, 47:1021–1037, 2011.

[56] Bernard Dumas and Elisa Luciano. An exact solution to a dynamic portfolio

choice problem under transactions costs. The Journal of Finance, 46(2):577–

595, 2012.

BIBLIOGRAPHY 168

[57] M.A. Duran and I.E. Grossmann. An outer-approximation algorithm for

a class of mixed-integer nonlinear programs. Mathematical Programming,

36:307–339, 1986.

[58] Samuel Eilon. Economic batch-size determination for multi-product schedul-

ing. OR, pages 217–227, 1959.

[59] Salah E Elmaghraby. The economic lot scheduling problem (ELSP): review

and extensions. Management Science, 24(6):587–598, 1978.

[60] Edwin J Elton and Martin J Gruber. The multi-period consumption invest-

ment problem and single period analysis. Oxford Economic Papers, pages

289–301, 1974.

[61] Edwin J Elton and Martin J Gruber. On the optimality of some multiperiod

portfolio selection criteria. Journal of Business, pages 231–243, 1974.

[62] Eugene F Fama. Multiperiod consumption-investment decisions. The Amer-

ican Economic Review, pages 163–174, 1970.

[63] James W Friedman. Oligopoly theory. CUP Archive, 1983.

[64] Chenpeng Fu, Ali Lari-Lavassani, and Xun Li. Dynamic mean–variance

portfolio selection with borrowing constraint. European Journal of Opera-

tional Research, 200(1):312–319, 2010.

BIBLIOGRAPHY 169

[65] Guillermo Gallego. An extension to the class of easy economic lot scheduling

problems. IIE Transactions, 22(2):189–190, 1990.

[66] Guillermo Gallego and Dong Xiao Shaw. Complexity of the ELSP with

general cyclic schedules. IIE Transactions, 29(2):109–113, 1997.

[67] Nicolae Gârleanu and Lasse Heje Pedersen. Dynamic trading with pre-

dictable returns and transaction costs. The Journal of Finance, 68(6):2309–

2340, 2013.

[68] Gerard Gennotte and Alan Jung. Investment strategies under transaction

costs: the finite horizon case. Management Science, 40(3):385–404, 1994.

[69] A.M. Geoffrion. Generalized benders decomposition. Journal of Optimiza-

tion Theory and Applications, 10(4):237–260, 1972.

[70] François Glineur et al. Computational experiments with a linear approxi-

mation of second-order cone optimization. Technical Report 0001, Faculte

Polytechnique de Mons, Mons, Belgium, 2000.

[71] Robert R Grauer and Nils H Hakansson. On the use of mean-variance and

quadratic approximations in implementing dynamic investment strategies:

A comparison of returns and investment policies. Management Science,

39(7):856–871, 1993.

BIBLIOGRAPHY 170

[72] N. Gülpınar, B. Rustem, and R. Settergren. Simulation and optimization

approaches to scenario tree generation. Journal of Economic Dynamics and

Control, 28(7):1291–1315, 2004.

[73] Nalan Gulpinar, Berc Rustem, and Reuben Settergren. Multistage stochas-

tic mean-variance portfolio analysis with transaction costs. In Innovations

in Financial and Economic Networks, pages 46–63. Edward Elgar Publishers

Ltd., UK, 2003.

[74] Nils H Hakansson. Optimal investment and consumption strategies under

risk for a class of utility functions. Econometrica: Journal of the Economet-

ric Society, pages 587–607, 1970.

[75] Nils H Hakansson. Capital growth and the mean-variance approach to port-

folio selection. Journal of Financial and Quantitative Analysis, 6(1):517–

557, 1971.

[76] Nils H Hakansson. On optimal myopic portfolio policies, with and without

serial correlation of yields. Journal of Business, pages 324–334, 1971.

[77] Nils H Hakansson. Convergence to isoelastic utility and policy in multiperiod

portfolio choice. Journal of Financial Economics, 1(3):201–224, 1974.

[78] Nils H Hakansson. Multi-period mean-variance analysis: Toward a general

theory of portfolio choice*. The Journal of Finance, 26(4):857–884, 2012.

BIBLIOGRAPHY 171

[79] Fred Hanssmann. Operations research in production and inventory control.

Wiley, 1962.

[80] Campbell R Harvey, John C Liechty, Merrill W Liechty, and Peter Müller.

Portfolio selection with higher moments. Quantitative Finance, 10(5):469–

485, 2010.

[81] Campbell R Harvey and Akhtar Siddique. Conditional skewness in asset

pricing tests. The Journal of Finance, 55(3):1263–1295, 2000.

[82] C. Helmberg, F. Rendl, R.J. Vanderbei, and H. Wolkowicz. An interior

point method for semidefinite programming. SIAM Journal on Optimiza-

tion, 6:342–361, 1996.

[83] Hassan Hijazi, Pierre Bonami, and Adam Ouorou. Robust delay-constrained

routing in telecommunications. Annals of Operations Research, 206(1):163–

181, 2013.

[84] Wen-Lian Hsu. On the general feasibility test of scheduling lot sizes for

several products on one machine. Management Science, 29(1):93–105, 1983.

[85] Norbert J Jobst, Michael D Horniman, Cormac A Lucas, and Gautam Mi-

tra. Computational aspects of alternative portfolio selection models in the

presence of discrete asset choice constraints. Quantitative Finance, 1(5):489–

501, 2001.

BIBLIOGRAPHY 172

[86] Phillip C Jones and Robert R Inman. When is the economic lot scheduling

problem easy? IIE transactions, 21(1):11–20, 1989.

[87] Hans Kellerer, Renata Mansini, and M Grazia Speranza. Selecting portfo-

lios with fixed costs and minimum transaction lots. Annals of Operations

Research, 99(1):287–304, 2000.

[88] Masakazu Kojima, Susumu Shindoh, and Shinji Hara. Interior-point meth-

ods for the monotone semidefinite linear complementarity problem in sym-

metric matrices. SIAM Journal on Optimization, 7(1):86–125, 1997.

[89] Hiroshi Konno and Annista Wijayanayake. Portfolio optimization problem

under concave transaction costs and minimal transaction unit constraints.

Mathematical Programming, 89(2):233–250, 2001.

[90] Georgios M Kopanos, Luis Puigjaner, and Michael C Georgiadis. Simulta-

neous production and logistics operations planning in semicontinuous food

industries. Omega, 40(5):634–650, 2012.

[91] Ailsa H Land and Alison G Doig. An automatic method of solving discrete

programming problems. Econometrica, 28(3):497–520, 1960.

[92] J.B. Lasserre. Global optimization with polynomials and the methods of

moments. SIAM Journal on Optimization, 1:796–817, 2001.

BIBLIOGRAPHY 173

[93] Larry J LeBlanc and David E Boyce. A bilevel programming algorithm

for exact solution of the network design problem with user-optimal flows.

Transportation Research Part B: Methodological, 20(3):259–265, 1986.

[94] D. Li and W.L. Ng. Optimal dynamic portfolio selection: Multiperiod mean-

variance formulation. Mathematical Finance, 10(3):387–406, 2000.

[95] Xun Li, Xun Yu Zhou, and Andrew EB Lim. Dynamic mean-variance port-

folio selection with no-shorting constraints. SIAM Journal on Control and

Optimization, 40(5):1540–1555, 2002.

[96] Andrew EB Lim and Xun Yu Zhou. Mean-variance portfolio selection with

random parameters in a complete market. Mathematics of Operations Re-

search, 27(1):101–120, 2002.

[97] Yeong-Cheng Liou and Jen-Chih Yao. Bilevel decision via variational in-

equalities. Computers & Mathematics with Applications, 49(7):1243–1253,

2005.

[98] Songsong Liu and Lazaros G Papageorgiou. Multiobjective optimisation of

production, distribution and capacity planning of global supply chains in

the process industry. Omega, 41(2):369–382, 2013.

[99] M.S. Lobo, M. Fazel, and S. Boyd. Portfolio optimization with linear and

BIBLIOGRAPHY 174

fixed transaction costs. Annals of Operations Research, 152(1):341–365,

2007.

[100] L. Lovasz and A. Schrijver. Cones of matrices and set-functions and 0-1

optimization. SIAM Journal on Optimization, 1(2):166–190, 1991.

[101] Ho-Yin Mak, Ying Rong, and Zuo-Jun Max Shen. Infrastructure plan-

ning for electric vehicles with battery swapping. Management Science,

59(7):1557–1575, 2013.

[102] Patrice Marcotte. Network design problem with congestion effects: A case

of bilevel programming. Mathematical Programming, 34(2):142–162, 1986.

[103] H. Markowitz. Portfolio selection. Journal of Finance, 7(1):77–91, 1952.

[104] H.M. Markowitz. Portfolio Selection: Efficient Diversification of Invest-

ments. Wiley, New York, 1959.

[105] William L Maxwell. The scheduling of economic lot sizes. Naval Research

Logistics Quarterly, 11(2):89–124, 1964.

[106] Athanasios Migdalas, Panos M Pardalos, and Peter Värbrand. Multilevel

optimization: algorithms and applications, volume 20. Springer, 1998.

[107] Dragan Miljkovic. Privatizing state farms in yugoslavia. Journal of Policy

Modeling, 24(2):169–179, 2002.

BIBLIOGRAPHY 175

[108] Tan Miller, Terry L Friesz, and Roger L Tobin. Heuristic algorithms for de-

livered price spatially competitive network facility location problems. Annals

of Operations Research, 34(1):177–202, 1992.

[109] Ciamac Moallemi and Mehmet Saglam. Dynamic portfolio choice with linear

rebalancing rules. Available at SSRN 2011605, 2012.

[110] Renato DC Monteiro. Primal–dual path-following algorithms for semidefi-

nite programming. SIAM Journal on Optimization, 7(3):663–678, 1997.

[111] I Moon, BC Giri, K Choi, et al. Economic lot scheduling problem with

imperfect production processes and setup times. Journal of the Operational

Research Society, 53(6):620–629, 2002.

[112] J. Mossin. Optimal multiperiod portfolio policies. The Journal of Business,

41(2):215–229, 1968.

[113] John M Mulvey and Hercules Vladimirou. Stochastic network programming

for financial planning problems. Management Science, 38(11):1642–1664,

1992.

[114] Yu E Nesterov and Michael J Todd. Self-scaled barriers and interior-point

methods for convex programming. Mathematics of Operations Research,

22(1):1–42, 1997.

BIBLIOGRAPHY 176

[115] Yu E Nesterov and Michael J Todd. Primal-dual interior-point methods for

self-scaled cones. SIAM Journal on Optimization, 8(2):324–364, 1998.

[116] Hayri Önal, Delima H Darmawan, and Sam H Johnson III. A multilevel

analysis of agricultural credit distribution in east java, indonesia. Computers

& Operations Research, 22(2):227–236, 1995.

[117] Gurobi Optimization. The Gurobi Optimizer 4.5.

[118] L. Ozsen, C.R. Coullard, and M.S. Daskin. Capacitated warehouse location

model with risk pooling. Naval Research Logistics, 55:295–312, 2008.

[119] G.C. Pflug. Scenario tree generation for multiperiod financial optimization

by optimal discretization. Mathematical Programming, 89(2):251–271, 2001.

[120] E.S. Phelps. The accumulation of risky capital: A sequential utility analysis.

Econometrica: Journal of the Econometric Society, pages 729–743, 1962.

[121] M. C. Pinar. Mixed-integer second-order cone programming for lower hedg-

ing of american contingent claims in incomplete markets. Optimization Let-

ters, pages 1–16, 2011. Online First.

[122] Jack Rogers. A computational approach to the economic lot scheduling

problem. Management Science, 4(3):264–291, 1958.

BIBLIOGRAPHY 177

[123] Paul A Samuelson. Lifetime portfolio selection by dynamic stochastic pro-

gramming. The Review of Economics and Statistics, 51(3):239–246, 1969.

[124] Z.J.M Shen, C.R. Coullard, and M.S. Daskin. A joint location-inventory

model. Transportation Science, 37:40–55, 2003.

[125] Hanif D Sherali, Allen L Soyster, and Frederic H Murphy. Stackelberg-

nash-cournot equilibria: characterizations and computations. Operations

Research, 31(2):253–276, 1983.

[126] H.D. Sherali and W.P. Adams. A hierarchy of relaxations between the con-

tinuous and convex hull representations of zero-one programming problems.

SIAM Journal of Discrete Mathematics, 3:411–430, 1990.

[127] H.D. Sherali and W.P. Adams. A hierarchy of relaxations for mixed-integer

zero-one programming problems. Discrete Applied Mathematics, 52:83–106,

1994.

[128] Policarpio Antonio Soberanis. Risk optimization with p-order cone con-

straints. PhD Dissertation. University of Iowa, 2009.

[129] M.C. Steinbach. Markowitz revisited: Mean-variance models in financial

portfolio analysis. Siam Review, 43(1):31–85, 2001.

[130] R.A. Stubbs and S. Mehrotra. A branch-and-cut method for 0-1 mixed

BIBLIOGRAPHY 178

convex programming. Mathematical Programming Series A, 86:515–532,

1999.

[131] Ik Sun Lee and SH Yoon. Coordinated scheduling of production and delivery

stages with stage-dependent inventory holding costs. Omega, 38(6):509–521,

2010.

[132] J.A. Taylor and F.S. Hover. Convex models of distribution and system

reconfiguration. IEEE Transactions on Power Systems, 27(3), August 2012.

[133] SA Torabi, B Karimi, and SMT Fatemi Ghomi. The common cycle economic

lot scheduling in flexible job shops: The finite horizon case. International

Journal of Production Economics, 97(1):52–65, 2005.

[134] Ann Van Ackere. The principal/agent paradigm: its relevance to various

functional fields. European Journal of Operational Research, 70(1):83–103,

1993.

[135] R.J. Vanderbei. LOQO user’s manual—version 3.10. Optimization Methods

and Software, 12:485–514, 1999.

[136] Luis N Vicente and Paul H Calamai. Bilevel and multilevel programming:

A bibliography review. Journal of Global Optimization, 5(3):291–306, 1994.

[137] J.P. Vielma, S. Ahmed, and G.L. Nemhauser. A lifted linear programming

BIBLIOGRAPHY 179

branch-and-bound algorithm for mixed-integer conic quadratic programs.

INFORMS Journal on Computing, 20(3):438–450, 2008.

[138] Heinrich Von Stackelberg. The theory of the market economy. William

Hodge, 1952.

[139] T. Westerlund and F. Pettersson. An extended cutting plane method for

solving convex MINLP problems. Computers and Chemical Engineering,

19:131–136, 1995.

[140] Robert L Winkler and Christopher B Barry. A bayesian model for portfolio

selection and revision. The Journal of Finance, 30(1):179–192, 2012.

[141] Atsushi Yoshimoto. The mean-variance approach to portfolio optimization

subject to transaction costs. Journal of the Operations Research Society of

Japan, 39(1):99–117, 1996.

[142] DZ Zhang et al. An integrated production and inventory model for a whole

manufacturing supply chain involving reverse logistics with finite horizon

period. Omega, 41(3):598–620, 2013.

[143] Xun Yu Zhou and G Yin. Markowitz’s mean-variance portfolio selection with

regime switching: A continuous-time model. SIAM Journal on Control and

Optimization, 42(4):1466–1482, 2003.

BIBLIOGRAPHY 180

[144] Paul H Zipkin. Computing optimal lot sizes in the economic lot scheduling

problem. Operations Research, 39(1):56–63, 1991.

advanced optimization and statistical methods in portfolio … · 2014-09-05 · this dissertation...

Documents